Coordinate Geometry. Positive gradients: Negative gradients:


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1 8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight lines 8:0 The gradient intercept form of a straight line: = m + c Investigation: What does = m + c tell us? 8:0 The equation of a straight line, given point and gradient 8:07 The equation of a straight line, given two points 8:08 Parallel and perpendicular lines 8:09 Graphing inequalities on the number plane Fun Spot: Wh did the banana go out with a fig? Mathemathical Terms, Diagnostic Test, Revision Assignment, Working Mathematicall Learning Outcomes Students will be able to: Find the distance between two points. Find the midpoint of an interval. Find the gradient of an interval. Graph straight lines on the Cartesian plane. Use the gradientintercept form of a straight line. Find the equation of a straight line given a point and the gradient, or two points on the line. Identif parallel and perpendicular lines. Graph linear inequalities on the Cartesian plane. Areas of Interaction Approaches to Learning (Knowledge Acquisition, Problem Solving, Communication, Logical Thinking, IT Skills, Reflection), Human Ingenuit 00
2 The French mathematician René Descartes first introduced the number plane. He realised that using two sets of lines to form a square grid allowed the position of a point in the plane to be recorded using a pair of numbers or coordinates. Coordinate geometr is a powerful mathematical technique that allows algebraic methods to be used in the solution of geometrical problems. In this chapter, we will look at the basic ideas of: the distance between two points on the number plane the midpoint of an interval gradient (or slope) the relationship between a straight line and its equation. We shall then see how these can be used to solve problems. 8:0 The Distance etween Two Points The number plane is the basis of coordinate geometr, an important branch of mathematics. In this chapter, we will look at some of the basic ideas of coordinate geometr and how the can be used to solve problems. prep qu A a = b + c b = a + c iz Which of the following is the correct statement of Pthagoras theorem for the triangle shown? c 8:0 For questions to, use Pthagoras theorem to find the value of d. cm d cm cm cm cm b C c = a + b a m dm d cm m CHAPTER 8 COORDINATE GEOMETRY 0
3 A Distance A 7 Distance A =... units. =... units Distance A =... units. A A Find the distance A. 0 Find the distance A. 0 A A A Distance A =... units. Pthagoras theorem can be used to find the distance between two points on the number plane. worked eamples Find the distance between the points (, ) and (, ). If A is (, ) and is (, ) find the length of A. Solutions 7 A(, ) C (, ) (, ) A(, ) 0 c = a + b c = a + b A = AC + C A = AC + C = + = + = + 9 = 9 + = = A = A = the length of A is units. the length of A is unit. C is a surd. We simplif surds if the are perfect squares. drawing a rightangled triangle we can use Pthagoras theorem to find the distance between an two points on the number plane. C A A C A C AC 0 INTERNATIONAL MATHEMATICS
4 Distance formula A formula for finding the distance between two points, A(, ) and (, ), can be found using Pthagoras theorem. We wish to find the length of interval A. Now LM = (since LM = MO LO) (ACML is a rectangle) and RS = (since RS = RO SO) Now (CSR is a rectangle) A = AC + C (Pthagoras theorem) = ( ) + ( ) A = AC = C = ( ) + ( ) R S (, ) C A(, ) O L M 0 The distance A between A(, ) and (, ) is given b: d = ( ) + ( ) worked eamples Find the distance between the Find the distance between the points (, 8) and (, ). points (, 0) and (8, ) Solutions Distance = ( ) + ( ) Distance = ( ) + ( ) (, ) = (, 8) and (, ) = (, ) (, ) = (, 0) and (, ) = (8, ) d = ( ) + ( 8) d = ( 8 ) + ( 0) = ( ) + ( ) = ( 0) + ( ) = + = 00 + = 0 = Distance 7 (using a calculator to answer to decimal places). Distance 8 (using a calculator to answer to decimal places). You should check that the formula will still give the same answer if the coordinates are named in the reverse wa. Hence, in eample, if we call (, ) = (, ) and (, ) = (, 8), we would produce the same answer. CHAPTER 8 COORDINATE GEOMETRY 0
5 Eercise 8:0 Use Pthagoras theorem to find the length of each of the following. (Leave our answer as a surd, where necessar.) a b 0 0 (7, 9) A(, ) 8 C(, ) (7, ) A(, ) C(7, ) Foundation Worksheet 8:0 Distance between points Use Pthagoras theorem to find the length of the hpotenuse in each of the following. a b 7 Find the distance A in each of the following. a b A A Find the length of A in each of the following. a b (, ) A(, ) A (, ) (, ) c d e (, ) A(, ) (, ) A(, ) 0 0 C 0 C A(, ) C (, ) Find the lengths C and AC and use these to find the lengths of A. (Leave our answers in surd form.) a (, ) b (, ) c ( 7, ) A(0, 0) C 0 C A(, ) C A(, ) d e f A(, ) (, ) 0 0 A(, ) (, ) C C 0 A(, ) (, ) C 0 INTERNATIONAL MATHEMATICS
6 Use Pthagoras theorem to find the length of interval A in each of the following. (Leave answers in surd form.) a A(, ) b c A(, ) A(, ) (, ) (, ) (, ) d e f A(, ) A(, ) (, ) 0 0 (, ) 0 (, ) A(8, ) 7 8 Use the formula d = ( ) + ( ) to find the distance between the points: a (, ) and (7, ) b (0, ) and (8, 7) c (, ) and (, ) d (, ) and (, ) e (, ) and (, 7) f (, 9) and (, ) g (, 0) and (, ) h (8, ) and (7, 0) i (, ) and (, ) j (, ) and ( 7, ) k (, ) and (, ) l (, ) and (, ) a Find the distance from the point (, ) to the origin. b Which of the points (, ) or (, ) is closer to the point (, 0)? c Find the distance from the point (, ) to the point (, ). d Which of the points (7, ) or (, ) is further from (0, 0)? Making a sketch will help. a The vertices of a triangle are A(0, 0), (, ) and C(, ). Find the length of each side. b ACD is a parallelogram where A is the point (, ), is (, ), C is (, ) and D is (, ). Show that the opposite sides of the parallelogram are equal. c Find the length of the two diagonals of the parallelogram in part b. d EFGH is a quadrilateral, where E is the point (0, ), F is (, ), G is (, ) and H is (, ). Prove that EFGH is a rhombus. (The sides of a rhombus are equal.) e (, ) is the centre of a circle. (, ) is a point on the circumference. What is the radius of the circle? f Prove that the triangle AC is isosceles if A is (, ), is (, ) and C is (, ). (Isosceles triangles have two sides equal.) g A is the point (, 7) and is (, ). M is halfwa between A and. How far is M from? CHAPTER 8 COORDINATE GEOMETRY 0
7 8:0 The Midpoint of an Interval prep quiz 8: What is the average of and 0? What is the average of and? What number is halfwa between and 0? What number is halfwa between and? What number is 9 What number is halfwa between halfwa between and? and? =? =? 0 0 The midpoint of an interval is the halfwa position. If M is the midpoint of A then it will be halfwa between A and. 8 7 A(, ) (0, 7) M(p, q) If M is the midpoint of A then AM = M. Consider the coordinates. Note: 7 is halfwa between and 0. Consider the coordinates. Note: is halfwa between and 7. The average of and 0 is 7. The average of and 7 is = = Formula: p + = Formula: q + = INTERNATIONAL MATHEMATICS
8 Midpoint formula M(p, q) (, ) M = + +, A(, ) Could ou please sa that in English, Miss? 0 The midpoint, M, of interval A, where A is (, ) and is (, ), is given b: M =, worked eamples Find the midpoint of the interval Find the midpoint of interval A, if A is joining (, ) and (8, 0). the point (, ) and is (, ). Solutions Midpoint = , Midpoint = , = , = , = (, 8) = ( ,  ) or ( ,  ) Eercise 8:0 Use the graph to find the midpoint of each interval. a b (, ) (, ) (, ) (, ) Foundation Worksheet 8:0 Midpoint Read the midpoint of the interval A from the graph Find the midpoint of the interval that joins: a (, ) to (0, 8) b... Find the midpoint of the interval that joins: a (, ) to (, ) b... CHAPTER 8 COORDINATE GEOMETRY 07
9 c (, ) d e (, ) (, ) (, ) (, ) (, ) Use the graph to find the midpoints of the intervals: a A b CD c GH d EF e LM f PQ g RS h TU i VW H 8 C U D T A 0 F G E Q L R S W M 8 P V Find the midpoint of each interval A if: a A is (, ), is (, 0) b A is (, 8), is (, ) c A is (, ), is (8, 7) d A is (0, 0), is (, ) e A is (, 0), is (, ) f A is (, ), is (, ) g A is ( 8, ), is (0, 0) h A is (, ), is (, ) i A is (, ), is (, 7) Find the midpoint of the interval joining: a (, ) and (, ) b (8, ) and (7, ) c (, ) and (, ) d (, 7) and ( 7, ) e (0, ) and (, 0) f (, ) and (, ) g (, 98) and (, ) h (8, ) and (7, 9) i (00, ) and (, 00) a i Find the midpoint of AC. ii Find the midpoint of D. iii Are the answers for i and ii the same? iv What propert of a rectangle does this result demonstrate? A(, ) (, ) D(, ) C(, ) 0 08 INTERNATIONAL MATHEMATICS
10 b If (, ) and (, 0) are points at opposite ends of a diameter of a circle, what are the coordinates of the centre? c i ii iii iv Find the midpoint of AC. Find the midpoint of D. Are the answers for i and ii the same? What propert of a parallelogram does this result demonstrate? 7 0 C(7, 7) (, ) D(, ) 7 A(, ) a If the midpoint of (, k) and (, ) is (8, ), find the value of k. b The midpoint of A is (7, ). Find the value of d and e if A is the point (d, 0) and is (, e). c The midpoint of A is (, ). If A is the point (, ), what are the coordinates of? d A circle with centre (, ) has a diameter of A. If A is the point (, ) what are the coordinates of? 7 8 a If A is the point (, ) and is the point (, 0), what are the coordinates of the points C, D and E? b If A is the point (, ) and D is the point (, 0), what are the coordinates of the points, C and E? E D C A a Use coordinate geometr to show that the points A(, 0), (8, 0), C(, ) and D(, ) form a parallelogram. b Use coordinate geometr to show that the points (, ), (, ), (, ) and (, 0) form a rectangle. Roger has different pizza toppings. How man different pizzas could be made using: topping? toppings? an number of toppings? CHAPTER 8 COORDINATE GEOMETRY 09
11 8:0 The Gradient of a Line prep quiz 8:0 0 F (, ) D (0, ) A C E G (, ) (, ) (7, ) (, ) (7, ) Which is steepest, A or EF? and If I travel from left to right, between which pairs of letters am I travelling upwards? and 7 etween which pairs of letters am I travelling downwards? Sa whether the hill is sloping up, down, or not at all, at the points 8 A 9 G 0 F The gradient or slope of a line is a measure of how steep it is. Negative gradient Positive gradient A higher positive gradient If we move from left to right the line going down is said to have a negative gradient (or slope). The line going up is said to have a positive gradient (or slope). If the line is horizontal (not going up or down) its gradient is zero. We find the gradient of a line b comparing its rise (change in ) with its run (change in ). Gradient rise = = run change in change in So a gradient of  means that for ever run of there is a rise of (or for ever that ou go across ou go up ). 0 INTERNATIONAL MATHEMATICS
12 Finding the gradient of a line Select an two points on the line. Join the points and form a rightangled triangle b drawing a vertical line from the higher point and a horizontal side from the lower point. Find the change in the coordinates (rise) and the change in the coordinates (run). Use the formula above to find the gradient. A run rise worked eamples Use the points A and to find the gradient of the line A in each case. (, ) (, ) (, ) A(, ) A(, ) A(, ) Solutions Gradient change in = change in up = across =  change in m = change in m = up = across = =  =  change in change in down across = m is used for gradient Gradient is generall left as an improper fraction instead of a mied numeral. So we write  instead of . Architectural design often requires an understanding of gradients (slopes). CHAPTER 8 COORDINATE GEOMETRY
13 Gradient formula We wish to find a formula for the gradient of a line A where A is (, ) and is (, ). A(, ) (, ) C(, ) 0 rise Gradient of A = run change in = change in C = AC m = (opposite sides of a rectangle are equal) m is used for gradient The gradient of the line that passes through the points A(, ) and (, ) is given b the formula: m = worked eamples Find the gradient of the straight line passing through the following points. (, ) and (, 7) (, ) and (, ) Solutions Let (, ) be (, ) Let (, ) be (, ) and (, ) be (, 7). and (, ) be (, ). Gradient = m = ( ) = = =  = The gradient is . The gradient is . It doesn't matter which point is called (, ). Eercise 8:0 For each of the following, state if the line has a positive or negative gradient. a b Foundation Worksheet 8:0 Gradients Find the gradient of the line from the graph. a b Find the lengths of the missing sides on each triangle and then find the gradient. a b (, ) (, ) (, 0) (, ) INTERNATIONAL MATHEMATICS
14 c d e Using Gradient = find the gradient of A in each of the following: Run a b c (, ) A(0, ) (, ) (, ) A(, ) A(, ) d A(, 9), (, 0) e A(0, ), (, 0) f A(, 8), (, 8) a b c Calculate the gradients of the four lines. Which lines have the same gradients? Which lines are parallel? E G D F On the same number plane, draw: a a line through (0, 0) with a gradient of b a line through (, ) which is parallel to the line in a. c Do the lines in a and b have the same gradient? 0 A H C If two lines have the same gradient the are parallel. Use the formula m = to find the gradient of the straight line passing through the points: a (, ) and (, 7) b (, ) and (, ) c (, ) and (7, ) d (0, 0) and (, ) If a line has no e (0, ) and (, ) f (, 0) and (, ) slope m = 0. g (, ) and (, ) h (7, 7) and (, ) i (9, ) and (, 7) j (, ) and (, ) k (, ) and (0, ) l (, ) and (, ) m (, ) and (, 9) n (, ) and (, ) o (, ) and (7, ) p (, ) and (, 7) q (, ) and (, ) r (, ) and (, ) CHAPTER 8 COORDINATE GEOMETRY
15 7 8 a b c d Find the gradient of the line that passes through A(, ) and (, ). Find the slope of the line that passes through O(0, 0) and (, ). On the graph shown, all of the points A,, C and D lie on the same straight line, + =. Find the gradient of the line using the points: i A and ii C and D iii A and D iv and C Conclusion: An two points on a straight line can be used to find the gradient of that line. A straight line has onl one gradient. Use the gradient of an interval to show that the points (, ), (, ) and (, ) are collinear (ie, lie on the same straight line). a i Find the gradient of C and of AD. ii Find the gradient of A and of DC. iii What kind of quadrilateral is ACD? Give a reason for our answer. b Prove that a quadrilateral that has vertices A(, ), (9, ), C(, 0) and D(, ) is a parallelogram. (It will be necessar to prove that opposite sides are parallel.) Use the fact that a rhombus is a parallelogram with a pair of adjacent sides equal to prove that the points A(, ), (, ), C(8, 8) and D(, ) form the vertices of a rhombus. A(0, ) (, ) C(, ) 0 C(, 0) 7 (, 7) C(7, 8) A(, ) D(, ) INTERNATIONAL MATHEMATICS
16 8:0 Graphing Straight Lines A straight line is made up of a set of points, each with its own pair of coordinates. Coordinate geometr uses an equation to describe the relationship between the  and coordinates of an point on the line. In the diagram, the equation of the line is + =. From the points shown, it is clear that the relationship is that the sum of each point s coordinates is. A point can onl lie on a line if its coordinates satisf the equation of the line. For the points (, ) and (, ), it is clear that the sum of the coordinates is not equal to. So the do not lie on the line. (, ) (, ) (, ) (0, ) (, ) + = (, ) (, ) The and in the equation are the point s coordinates. 0 (, ) + = (, ) To graph a straight line we need: an equation to allow us to calculate the  and coordinates for each point on the line a table to store at least two sets of coordinates a number plane on which to plot the points. Two important points on a line are: the intercept (where the line crosses the ais) This is found b substituting = 0 into the line s equation and then solving for. the intercept (where the line crosses the ais) This is found b substituting = 0 into the line s equation and then solving for. Is THAT all? He, no problem! I can do that! intercept intercept CHAPTER 8 COORDINATE GEOMETRY
17 Horizontal and vertical lines The line shown on the graph on the right is vertical. elow, we have put the points on the line into a table. 0 There seems to be no connection between and. However, is alwas. So the equation is =. = 0 Vertical lines have equations of the form = a where a is where the line cuts the ais. This line is =. 0 This line is =. The cut the ais at and. The line on the right is horizontal. elow, we have put the points on the line, into a table. 0 There seems to be no connection between and. However, is alwas. So the equation is =. 0 Horizontal lines have equations of the form = b where b is where the line cuts the ais. This line is =. 0 This line is =. The cut the ais at and. INTERNATIONAL MATHEMATICS
18 worked eamples Draw the graph of each straight line. = + = + = + = 0 Solutions = 0 + = 0 + = when = 0, = 0 = when =, = = when =, = = when = 0, 0 + = 0 + = = when =, + = + = = when =, + = + = = when = 0, 0 + = 0 + = = when =, + = + = = = when =, + = + = = = Since it is difficult to plot fractions, an intercept method would be better here: At the ais, = = = = At the ais, = = = = continued CHAPTER 8 COORDINATE GEOMETRY 7
19 + = 0 It can be seen that if a table of values is used here fractions will result. The intercept method is better again. Rewrite the equation as = At the ais, = 0 0 = = = At the ais, = = = = 0 Eercise 8:0 Using separate aes labelled from to, draw the graph of the following lines using an of the above methods. a = b = c + = d + = Graph the lines represented b these equations using an appropriate method. a + = b = c = + d = 0 e = 0 f = g = + h = On which of the following lines does the point (, ) lie? a = + b + + = 0 c = 8 d = e = 0 f = 0 The line + = 0 passes through which of the following points? a (, ) b (7, ) c (, ) d (, 0) e (8, ) f (, 0) For each number plane, write down the equations of the lines A to F. a A C b A C D D E E 0 F 0 F Foundation Worksheet 8:0 Graphing lines Complete the tables for the equations: a = b = + c + = Read off the  and intercept of line: A a A b 0 8 INTERNATIONAL MATHEMATICS
20 7 Using values from to on each ais, draw the graphs of the following straight lines. Use a new diagram for each part. a =, =, =, = 0 b =, = 0, =, = c =, =, =, = d =, =, =, = e =, = 0, = 0, = Which of these encloses a square region? Match each of the graphs A to F with one of the following equations: = = + = 0 = + = + = A C D E F 8 Which of the lines A,, C or D could be described b the following equation. a = b + = c + + = 0 d + = 0 A 9 0 Use the intercept method to graph the following lines. a + = b + = c + = d = e = f = Draw the graph of each equation. + a = b = c = D d + = 7 e = 0 f = 0 C CHAPTER 8 COORDINATE GEOMETRY 9
21 8:0 The Gradient Intercept Form of a Straight Line: = m + c prep quiz 8:0 If = 0, what is the value of: m + c If = 0, what is the value of when: = + = What is the gradient of: line A line What are the coordinates of the intercept of: 7 line A 8 line 9 Does ever point on the ais have an coordinate of 0? 0 Can the intercept of a line be found b putting = 0. A As ou learnt in ook : The equation of a line can be written in several was. For instance, = 0, = and = are different was of writing the same equation. When the equation is written in the form = 0, it is said to be in general form. The form = is a particularl useful wa of writing the equation of a line. It allows us to get information about the line directl from the equation. investigation 8:0 Investigation 8:0 What does = m + c tell us? Please use the Assessment Grid on the page to help ou understand what is required for this Investigation. For each of the following equations complete the table b: graphing each line on a Cartesian grid calculating the gradient of each line (use an two points that the line passes through), and noting where the graph crosses the ais (intercept). Line Equation Gradient intercept = = =  + = = is the coefficient of. = 0 INTERNATIONAL MATHEMATICS
22 Are there an patterns ou can find connecting the gradients and the intercepts and the numbers in the equations of the lines? Eplain these patterns or connections. Tr to write a rule that will help ou write down the gradient and intercept for a line without drawing the line. Using our rule, write down the equations of the lines shown in the Cartesian grid below. 0 8 A D F 8 E G making a table of values for our equations, check that our equations represent the lines given. Do ou notice anthing special about the lines that have the same colour? 7 Can ou discuss some possible real life applications of straight lines and their equations? When an equation of a line is written in the form = m + c, m gives the gradient of the line and c gives the intercept of the line. Clearl, lines with the same gradient are parallel. When an equation of a line is written in the form a + b + c = 0, where a, b and c are integers and a > 0, it is said to be in general form. CHAPTER 8 COORDINATE GEOMETRY
23 Assessment Grid for Investigation 8:0 What does = m + c tell us? The following is a sample assessment grid for this investigation. You should carefull read the criteria before beginning the investigation so that ou know what is required. Assessment Criteria (, C, D) for this investigation Achieved a None of the following descriptors has been achieved. 0 b Some help was needed to complete the table and identif the simple patterns in questions and. Criterion Investigating Patterns c d e Mathematical problemsolving techniques have been selected and applied to accuratel graph the lines required and complete the table, with some suggestion of emerging patterns. The student has graphed the required lines and used the patterns evident in the table to find a connecting rule to give the equations of the lines in question. The patterns evident between the equations and their graphs have been eplained and summarised as a mathematical rule. The patterns for the lines in part have been eplained. 7 8 Criterion C Communication in Mathematics a None of the following descriptors has been achieved. 0 b c d There is a basic use of mathematical language and representation. Lines of reasoning are insufficient. There is satisfactor use of mathematical language and representation. Graphs, tables and eplanations are clear but not alwas logical or complete. A good use of mathematical language and representation. Graphs are accurate, to scale and full labelled. Eplanations and answers are complete and concise. a None of the following descriptors has been achieved. 0 Criterion D Reflection in Mathematics b c d An attempt has been made to eplain whether the results make sense, with connection to possible reallife applications. There is a correct but brief eplanation of whether results make sense and how the were found. A description of the important aspects of the graphs is given and the relation to their equations. There is a critical eplanation of the graphs obtained and their related equations. All results are full eplained and justified, and specific reference to reallife applications of lines is given. INTERNATIONAL MATHEMATICS
24 worked eamples Write down the gradient and intercepts of these lines. a = b = c + = = + =  + m = b = b = m = Gradient = Gradient = intercept = intercept = From the graph of the line shown, write down the gradient and intercept of the line and hence its equation. From the graph: The intercept (c) = The gradient (m) =  = the equation = m + c becomes = Use the intercept and the gradient to graph the following lines: a = 0 0 m =  b = Gradient =  intercept = This line is rising, so,the gradient is positive. Start at the intercept of (c) =. Rise Gradient = = =  and is positive so for ever Run unit across there is a rise of units. b + = First, rearrange the equation in the form = m + c. + = = + = Start at the intercept (c) = Rise Gradient = =  Run and is negative so for ever units across there is a fall of units. Remember! A negative gradient alwas slopes down to the right. CHAPTER 8 COORDINATE GEOMETRY
25 Eercise 8:0 Write the gradient and intercept of the following lines: a = b = + c = 9 7 d = +  e = first rearranging the equation in the form = m + c write the gradient and intercept of the following lines: a = + 8 b = c + = 0 d = e = f + 0 = 0 g = h = i = 0 j = Use the graph to find the gradient and intercept of each line and hence write the equation of each line in the general form. (The general form of a line is a + b + c = 0, where a, b and c are whole numbers and a is positive.) C Foundation Worksheet 8:0 Gradient intercept form For each line find from the graph its: a intercept b gradient What is = m + b when: a m = and b = b m = and b = Find the equation of a line with: a a gradient of and intercept of A D Draw the graphs of the following using onl the gradient and intercept (follow eample ). a = b = c = d = 8 e + = 0 f + + = 0 g + = 0 h = E 8:0 Equation grapher Slope is another name for gradient. INTERNATIONAL MATHEMATICS
26 8:0 The Equation of a Straight Line, Given Point and Gradient A line has the equation = +. Find: its gradient its intercept the value of when = the value of when =. If = m + c, find the value of c if: m =, = 0, = m =, =, = 7 m =, =, = 8 m =, =, = 0 9 m = 0, =, = 0 m =, =, = prep quiz 8:0 Through a given point, an number of lines can be drawn. Each of these lines has a different gradient, so the equation of a straight line can be found if we know its gradient and a point through which it passes. The equation of a line through (, ) having a gradient of can be found b beginning with the formula = m + b. We know that m =, but we must also find the value of b. To do this, we substitute the coordinates of (, ) into the equation, as a point on the line must satisf its equation. Working: = m + c (formula) = + c (m = is given) = () + c [(, ) lies on the line] = + c c = The equation of the line is =. 0 m = (, ) m = m = m = To find the equation of a straight line that has a gradient of and passes through (7, ): () Substitute m =, = 7 and = into the formula = m + c to find the value of c. () Rewrite = m + c replacing m and c with their numerical values. We can use the method above to discover a formula that could be used instead. Question: What is the equation of a line that has a gradient m and passes through the point (, )? Working: = m + c (formula; gradient is m) (, ) = m + c [(, ) lies on the line] c = m The equation of the line is = m + ( m ) 0 ie = m m = m( ) This last form of the answer is the easiest to remember as it could be written as = m. gradient = m CHAPTER 8 COORDINATE GEOMETRY
27 The equation of a line with gradient m, that passes through the point (, ) is given b: = m( ) or = m. worked eamples Find the equation of the line that passes through (, ) and has gradient. A straight line has gradient  and passes through the point (, ). Find the equation of this line. You can use either formula. = m( ) or = m + c Solutions Let the equation of the line be: or = m( ) = m + c (, ) is (, ), m = = + c (m = is given) = ( ) = () + c [(, ) lies on the line] = = + c = + is the c = equation of the line. The equation is = +. Let the equation be: or = m( ) = m + c (, ) is (, ), m = =  + c (m =  is given) =  ( ) =  () + c [(, ) is on the line] =  + =  + c = is the c =  The equation is = equation of the line. Eercise 8:0 For each part, find c if the given point lies on the given line. a (, ), = + c c (, 0), = + c c (, ), = + c d (, ), = + c e (, ), = + c f (, 9), = + c Foundation Worksheet 8:0 Point gradient form Use = m( ) to find the equation of a line when: m =, (, ) = (, ) m =, (, ) = (, ) Find the equation of the straight line (giving answers in the form = m + c) if it has: a gradient and passes through the point (, ) b gradient and passes through the point (0, 0) c gradient and passes through the point (, ) d slope and passes through the point (, ) INTERNATIONAL MATHEMATICS
28 e f g gradient and passes through the point (, 8) gradient and passes through the point (0, 7) slope and passes through the point (, 0) h gradient  and passes through the point (, ) i gradient  and passes through the point (,  ) j slope  and passes through the point (, ) Slope is another name for gradient. a A straight line has a gradient of and passes through the point (, ). Find the equation of the line. b A straight line has a gradient of. If the line passes through the point (, ), find the equation of the line. c What is the equation of a straight line that passes through the point (, 0) and has a gradient of? d A straight line that passes through the point (, ) has a gradient of. What is the equation of this line? e A straight line that has a gradient of passes through the origin. What is the equation of this line? f Find the equation of the straight line that has a gradient of and passes through the point (, ). g (, 8) is on a line that has a gradient of. Find the equation of this line. h The point (, ) lies on a straight line that has a gradient of. What is the equation of this line? i Find the equation of the straight line that has a gradient of and passes through the midpoint of the interval joining (, ) and (, ). j A straight line passes through the midpoint of the interval joining (0, 0) and (, ). Find the equation of the line if its gradient is . 8:07 The Equation of a Straight Line, Given Two Points Onl one straight line can be drawn through two points. Given two points on a straight line, we can alwas find the equation of that line. Consider the line passing through (, ) and (, ). Let the equation of the line be: = m + c (formula) CHAPTER 8 COORDINATE GEOMETRY 7
29 First find the gradient using the two points. m = (, ) = (, ) ( =, ) = (, ) = = + c (since m = ) = ( + c) [(, ) lies on the line] c = The equation of the line is =. (, ) (, ) 0 To find the equation of a straight line that passes through the two points (, ) and (, ): Find the value of the gradient m, using the given points. For = m + c, find the value of c b substituting the value of m and the coordinates of one of the given points. Rewrite = m + c replacing m and c with their numerical values. Another method is to use the formula: = ( ) where (, ) and (, ) are points on the line. worked eample Find the equation of the line that passes through the points (, ) and (, 8). Solution Let the equation of the line be: = m + c Now m = = ( ) =  m = = + c (since m = ) (, 8) lies on the line. 8 = () + c c = The equation is = +. or (, ) = (, ) (, ) = (, 8) = ( ) (, ) is (, ), (, ) is (, 8) = [ ( ) ( )] =  ( + ) = ( + ) = + = + is the equation of the line. 8 INTERNATIONAL MATHEMATICS
30 Eercise 8:07 Find the gradient of the line that passes through the points: a (, 0) and (, ) b (, ) and (, ) c (, ) and (, ) d (, ) and (0, 9) e (, ) and (, ) f (, ) and (, ) g (0, 0) and (, ) h (, ) and (, ) i (, 8) and (, ) j (0, 0) and (, ) Use our answers for question to find the equations of the lines passing through the pairs of points in question. a Find the equation of the line that passes through the points (, ) and (, ). b The points A(, ) and (, 0) lie on the line A. What is the equation of A? c What is the equation of the line A if A is the point (, ) and is (, )? d Find the equation of the line that passes through the points (, ) and (, 8). substitution in this equation, show that (, 0) also lies on this line. e What is the equation of the line CD if C is the point (, ) and D is the point (, )? A is the point (, ), is the point (, ) and C is the point (, ). a Find the gradient of each side of AC. b Find the equation of each of the lines A, C and AC. c Find the intercept of each of the lines A, C and AC. d Find the equation of the line passing through point A and the midpoint of interval C. e Find the gradient and intercept of the line passing through point A and the midpoint of interval C. a b Find the equation of the line joining A(, ) and (, ). Hence show that C(, ) also lies on this line. A(, ), (, ) and C(, 8) are points on the number plane. Show that the are collinear. c Show that the points (, ), (, ) and (, 7) are collinear. Find the equation of the lines in general form that pass through the points: a (, ) and (, ) b (, ) and (, ) c (, ) and (, 7 ) d ( ,  ) and ( ,  ) m = A Collinear points lie on the same straight line. 0 C Recipe for question a. Find the equation of A. Substitute C into this equation. CHAPTER 8 COORDINATE GEOMETRY 9
31 8:08 Parallel and Perpendicular Lines prep quiz 8:08 D A 0 C D E A 0 F C In the diagram, A is perpendicular to C, and DE is perpendicular to EF. What is the gradient of each line? Find the gradient of A. Call this m. Are the lines parallel? Find the gradient of C. Call this m. If EF was drawn parallel to A, 7 Using our answers to and, find what would its gradient be? the product of the gradients, m m. Is it possible for two lines with 8 Find the gradient of DE. Call this m. with different gradients to be 9 Find the gradient of EF. Call this m. parallel? 0 Using our answers to 8 and 9, find the product of the gradients, m m. Questions to in the Prep Quiz remind us that: two straight lines are parallel if their gradients are equal the gradients of two lines are equal if the lines are parallel. Questions to 0 of the Prep Quiz suggest that a condition for two lines to be perpendicular might be that the product of their gradients is equal to. We do not intend to prove this here, but let us look at several pairs of lines where the product of the gradient is to see if the angle between the lines is 90. A Line r has gradient. m = Line s has gradient. m = Note that m m =. Line t has gradient . m =  Line u has gradient. m = Note that m m =. C Line v has gradient . m =  Line w has gradient . m =  Note that m m =. measurement, or use of Pthagoras theorem, we can show that the angle between each pair of lines is 90. If two lines are perpendicular, then the product of their gradients is. m m = (where neither gradient is zero). v w 0 r s u t 0 INTERNATIONAL MATHEMATICS
32 If the product of the gradients of two lines is, then the lines are perpendicular. Two lines with gradients of m and m are: parallel if m = m perpendicular if m m = (or m = ) where neither m nor m can equal zero. m worked eamples Which of the lines =, = + and = is perpendicular to + + = 0? Find the equation of the line that passes through the point (, ) and is perpendicular to =. Find the equation of the line that passes through the point (, ) and is parallel to =. Solutions Step : Find the gradient of + + = 0. Writing this in gradient form gives: =  The gradient of this line is . Step : Find the gradients of the other lines. The gradient of = is. The gradient of = + is. The gradient of = is. Step : Find which gradient in step will multipl  to give. Conclusion:  = + + = 0 is perpendicular to =. If m m =, then m = m Let the equation of the line be = m + b. Now the gradient of = is. m =  (since  = ) =  + c =  () + c [since (, ) lies on line] =  + c c =  The equation of the line is = Let the equation of the line be = m + c. = has gradient m = (Parallel lines have equal gradients.) = + c = () + c, [(, ) lies on the line] c = The equation of the line is = +. CHAPTER 8 COORDINATE GEOMETRY
33 Eercise 8:08 Are the following pairs of lines parallel or not? a = + and = b = and = c = + 7 and = + d = and = + e = + and = f = + and + = 0 g + = 0 and + + = 0 h + = and + = 8 Foundation Worksheet 8:08 Parallel and perpendicular lines Use = m + c to find the slope of the lines in columns A and. Which lines in columns A and are: a parallel? b perpendicular? Are the following pairs of lines perpendicular or not? a =  +, = + b =, = c =, =  + d =  +, =  e =, =  f = , =  g =, + + = 0 h + =, = 0 a Which of the following lines are parallel to = +? = + + = 0 = + = b Two of the following lines are parallel. Which are the? = + = = = = + 8 c A is the point (, ), is (, ), C is (, 7) and D is (8, 8). Which of the lines A, C, CD and DA are parallel? a Which of the following lines are perpendicular to =? = = + = = 0 + b Two of the following lines are perpendicular. Which are the? =  + =  =  c A is the point (, ), is the point (, ) and C is (, ). Prove that A C. a Find the equation of the line that has intercept and is parallel to =. b Line A is parallel to =. Find the equation of A if its intercept is. c d Line EF is parallel to = +. Its intercept is. What is the equation of EF? A line has a intercept of 0 and is parallel to the line + =. What is the equation of this line? a Find the equation of the line that has intercept and is perpendicular to = b The line A is perpendicular to = +. Its intercept is. What is the equation of A? c Find the equation of CD if CD is perpendicular to the line =  and has a intercept of 0. d A line has a intercept of and is perpendicular to the line = +. Find the equation of the line. INTERNATIONAL MATHEMATICS
34 7 8 a A is a line which passes through the point (, ). What is the equation of A if it is parallel to = +? b Find the equation of the line that passes through (, 0) and is parallel to =. c d e a A is the point (0, 0) and is the point (, ). Find the equation of the line that has intercept and is parallel to A. Find the equation of the line that has intercept and is parallel to the ais. What is the equation of the line that is parallel to the ais and passes through the point (, )? If A passes through the point (, ) and is perpendicular to = 7, find the equation of A in general form. b Find the equation of the line that passes through (, 0) and is perpendicular to =. Write our answer in general form. c d e A is the point (0, 0) and is the point (, ). Find the equation of the line that has intercept and is perpendicular to A. Give the answer in general form. Find the equation of the line that has intercept and is perpendicular to the ais. What is the equation of a line that is perpendicular to the ais and passes through (, )? 9 a b c d Find the equation of the line that is parallel to the line + = 0 and passes through the point (, ). Give the answer in general form. A line is drawn through (, ), perpendicular to the line + = 0. Find its equation in general form. A line is drawn through the point (, ), parallel to the line + 9 = 0. Where will it cross the ais? A line is drawn parallel to + = 0, through the points (, ) and (, a). What is the value of a? 0 In the diagram, the line + + = 0 cuts the ais and ais at E and C respectivel. D is the line =, A is parallel to the ais and E and CD are perpendicular to AC. Find the coordinates of the points A,, C, D and E. A E D C CHAPTER 8 COORDINATE GEOMETRY
35 8:09 Graphing Inequalities on the Number Plane prep quiz 8:09 For each number line graph, write down the appropriate equation or inequation On a number line, draw the graph of < where is a real number. In Prep Quiz 8:09 questions, and, we see that, once = is graphed on the number line, all points satisfing the inequation > lie on one side of the point and all points satisfing the inequation < lie on the other side. On the number plane, all points satisfing the equation = + lie on one straight line. All points satisfing the inequation < + will lie on one side of the line. All points satisfing the inequation > + will lie on the other side of the line. A = + < + Note: Inequations, C and D are often called half planes. In D, the line is part of the solution set. In and C, the line acts as a boundar onl, and so is shown as 0 0 a broken line. Choose points at random in each of the half planes in, C and D to confirm that all points in each half plane satisf the appropriate C > + D + inequation. 0 INTERNATIONAL MATHEMATICS 0 worked eamples Graph the region + > on the number plane. Graph a the union and b the intersection of the half planes representing the solutions of + and <. continued Points that lie on broken lines are not part of the solution.
36 Solutions Graph the boundar line + = as a broken line since it is not part of + >. + = 0 0 Discover which half plane satisfies the inequation + > b substituting a point from each side of the boundar into + >. (0, 0) is obviousl to the left of + =. substitute (0, 0) into + >. (0) + (0) >, which is false. (0, 0) does not lie in the half plane. (, ) is obviousl to the right of + =. substitute (, ) into + >. () + () >, which is true. (, ) lies in the half plane + >. Shade in the half plane on the (, ) side. Points that lie on broken lines are not part of the solution. + = (, ) 0 (0, 0) + > 0 Graph the two half planes using the method above. + = = Points above the boundar line satisf Points to the right of the boundar satisf +. < continued CHAPTER 8 COORDINATE GEOMETRY
37 a The union of the two half planes b The intersection is the region that is the region that is part of one or belongs to both half planes. It is the the other or both graphs. part that the graphs have in common. 0 0 The union is written: The intersection is written: {(, ): + < } {(, ): + < } Note: Initiall draw the boundar lines as broken lines. Part of each region has a part of the boundar broken and a part unbroken. Eercise 8:09 testing a point from each side of the line, write down the inequation for each solution set graphed below. a b = = Foundation Worksheet 8:09 Graphing inequalities Write down the inequalit that describes each region. a b Graph the inequalities: a < b = 0 0 c d e = + + = = INTERNATIONAL MATHEMATICS
38 = = On separate number plane diagrams, draw the graph of: a b > c < d + e < + f g + 0 h > 0 Draw a separate diagram to show the part of the number plane where: a 0 and 0, ie {(, ): 0 0} b 0 and 0, ie {(, ): 0 0} c 0 and/or 0, ie {(, ): 0 0} d 0 and/or 0, ie {(, ): 0 0} means intersection; means union. Use question of this eercise to sketch the region described b the intersection of: a < and b and < c + and d + and < e + and + f + and < Use question of this eercise to sketch the region described b the union of: a < and b + and < c + and d + and < {(, ):...} means: the set of points such that... Describe, in terms of the union or intersection of inequations, the regions drawn below. a b c = = = 0 = 7 Sketch the regions described below. a the intersection of and + b the union of < and < c the intersection of < + and + < 0 d the union of and < Fill in onl that part of the boundar which is part of the answer. 8 Graph the regions which satisf all of the following inequalities: a,, 0, b 0,, + CHAPTER 8 COORDINATE GEOMETRY 7
39 9 Write down the inequalities that describe each region. a b c = = = fun spot 8:09 Fun Spot 8:09 Wh did the banana go out with a fig? Answer each question and put the letter for that question in the bo above the correct answer. For the points A(, ), (, ), C(, ), D(0, ), find: E distance AC E distance CD E distance D A slope of A A slope of AD T slope of C T midpoint of DC N midpoint of AC I midpoint of A What is the gradient of the following lines: C = U = G + + = 0 What is the intercept of the following lines. D = T = L + = 0 Find the equation of the line with: C gradient of and a intercept of O slope of and a intercept of T intercept of and a slope of S intercept of and a slope of Write in the form = m + b: = 0 E + = 0 A =  In the diagram find: C (, ) D slope of A U coordinates of C A(, ) = = + =  (, ) (, ) = = +  (, ) (, )  = + = INTERNATIONAL MATHEMATICS
40 terms 8 mathematical Mathematical Terms 8 coordinates A pair of numbers that gives the position of a point in a number plane relative to the origin. The first of the coordinates is the coordinate. It tells how far right (or left) the point is from the origin. The second of the coordinates is called the coordinate. It tells how far the point is above (or below) the origin. distance formula Gives the distance between the points (, ) and (, ). d = ( ) + ( ) general form A wa of writing the equation of a line. The equation is written in the form a + b + c = 0. where a, b, c are integers and a > 0. gradient The slope of a line or interval. It can be measured using the formula: Gradient = rise run gradient formula Gives the gradient of the interval joining (, ) to (, ). m = gradient intercept form A wa of writing the equation of a line. eg =, =  + Run Rise When an equation is rearranged and written in the form = m + c then m is the gradient and c is the intercept. graph (a line) All the points on a line. To plot the points that lie on a line. interval The part of a line between two points. midpoint Point marking the middle of an interval. midpoint formula Gives the midpoint of the interval joining (, ) to (, ). Midpoint = , number plane A rectangular grid that allows the C(, ) A(, ) position of points in a plane to be 0 identified b E(0, ) an ordered (, ) pair of D(, ) numbers. origin The point where the ais and ais intersect, (0, 0). See under number plane. plot To mark the position of a point on the number plane. ais The horizontal number line in a number plane. See under number plane. intercept The point where a line crosses the ais. ais The vertical number line in a number plane. See under number plane. intercept The point where a line crosses the ais. Mathematical terms 8 CHAPTER 8 COORDINATE GEOMETRY 9
41 diagnostic test 8 Diagnostic Test 8: Coordinate Geometr These questions reflect the important skills introduced in this chapter. Errors made will indicate areas of weakness. Each weakness should be treated b going back to the section listed. Find the length of the interval A in each of the following. (Leave answers in surd form.) a b c A 0 A A Section 8:0 Use the distance formula to find the distance between the points: a (, ) and (7, 0) b (, 0) and (, ) c (, ) and (, ) Find the midpoint of the interval joining: a (, ) and (7, 0) b (, 0) and (, ) c (, ) and (, ) What is the gradient of each line? a b c Find the gradient of the line that passes through: a (, ), (, 7) b (, 8), (, ) c (0, ), (, ) a Does the point (, ) lie on the line + =? b Does the point (, ) lie on the line = +? c Does the point (, ) lie on the line =? 7 Graph the lines: a = + b = c + = 8 State the  and intercepts of the lines: a = b + = c + = 8:0 8:0 8:0 8:0 8:0 8:0 8:0 0 INTERNATIONAL MATHEMATICS
42 9 Graph the lines: a = b = c = 0 Write down the equation of the line which has: a a gradient of and a intercept of b a gradient of  and a intercept of c a intercept of and a gradient of Write each of the answers to question 0 in general form. What is the gradient and intercept of the lines: a = +? b =? c = +? Rearrange these equations into gradient intercept form. a + = 0 b + = 0 c + + = 0 Find the equation of the line that: a passes through (, ) and has a gradient of b has a gradient of and passes through (, ) c has a gradient of  and passes through (, 0) Find the equation of the line that: a passes through the points (, ) and (, ) b passes through the points (, ) and (, ) c passes through the origin and (, ) Find the equation of the line that: a has a intercept of and is parallel to = b passes through (, 7) and is parallel to = + c is perpendicular to =  + and passes through (, ) d is perpendicular to = and passes through (, ) 7 Write down the inequation for each region. a b c = Graph a the union and b the intersection of the half planes representing the solutions of + and <. = 0 + = 0 8:0 8:0 8:0 8:0 8:0 8:0 8:07 8:08 8:09 8:09 CHAPTER 8 COORDINATE GEOMETRY
43 assignment 8A Chapter 8 Revision Assignment Find: a the length A as a surd b the slope of A c the midpoint of A. (, ) A(, ) A is the point (, ) and is the point (7, 7). a What is the length A (as a surd)? b What is the slope of A? c What is the midpoint of A? A is the point (, ) and is the point (, ). a What is the equation of the line A? b The line A passes through the point (00, b). What is the value of b? c AC is perpendicular to A. Find its equation in general form. a A line has an intercept of and a gradient of. Find where the line crosses the ais and hence write down its equation. b c A line has a slope of  and a intercept of. What is its equation? What is its intercept? A line has an intercept of and a intercept of. What is its equation? The points X(, ), Y(, ) and Z(, 0) form a triangle. Show that the triangle is both isosceles and rightangled. A line is drawn perpendicular to the line + = 0 through its intercept. What is the equation of the line? Give the answer in general form. 7 A median of a triangle is a line drawn from a verte to the midpoint of the opposite side. Find the equation of the median through A of the triangle formed b the points A(, ), (, ) and C(, 8). 8 What inequalities describe the region shown? (, ) and intercept and graphs Using = m + c to find the gradient General form of a line Parallel and perpendicular lines Inequalities and regions Linear graphs and equations The coordinate sstem for locating points on the earth is based on circles. INTERNATIONAL MATHEMATICS
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