Chapter Thirteen. Coordinate Geometry USE GRAPH PAPER FOR HOMEWORK IN THIS CHAPTER, AS NECESSARY!

Transcription

1 Chapter Thirteen Coordinate Geometry USE GRAPH PAPER FOR HOMEWORK IN THIS CHAPTER, AS NECESSARY! Make sure you properly annotate all coordinate graphs in this chapter!

2 Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates and theorems in this chapter. D. Understand and apply the distance and midpoint formulas. E. Calculate and use the slopes of lines. F. Perform the basics of vector mathematics. G. Graph linear equations H. Write the equations of straight line graphs I. Organize and write a coordinate proof.

3 Section 13-1 The Distance Formula Homework Pages : 1-24, 28, 36 Exclude 14, 16

4 Objectives A. Understand and apply the terms origin, axes, quadrants, and coordinate plane. B. Properly graph points, lines, and circles on a coordinate plane. C. Derive and utilize the distance formula. D. Understand the components of the equation of a circle. E. Graph a circle in a coordinate plane based on its equation.

5 The Coordinate Plane coordinate plane: plane formed by the intersection of a horizontal and a vertical real number line, called the coordinate axes, where every point in the plane can be represented by an ordered pair of real numbers, called its coordinates. x-axis: the horizontal coordinate axis y-axis: the vertical coordinate axis origin: intersection of the coordinate axes

6 Coordinate Plane y-axis 5 (8, 5) origin x-axis

7 The Coordinate Plane - Quadrants Quadrants: four regions of the coordinate plane created by the two axes. y Quadrant II: x is negative; y is positive Quadrant I: Both x & y are positive x Quadrant III: Both x & y are negative Quadrant IV: x is positive; y is negative

8 Building the Distance Formula 1. Label the points. 2. Label the coordinates 3. Substitute values P 1 =(x 1,y 1 ) P 2 =(x 2,y 2 ) P 1 =(3,2) P 2 =(6,3) P 2 (xp 1,y 1 ) (x,y ) 2 2 y 2 y Distance between points on x-axis, x 2 x 1

9 Building the Distance Formula c 2 c a 2 a 2 b 2 b 2 (6,3) c 2 x x y y1 y 2 y 1 (3,2) x 2 x 1

10 Theorem 13-1 (Distance Formula) The distance d between points (x 1, y 1 ) and (x 2, y 2 ) is given by: d x x y y (x 2, y 2 ) d y 2 - y 1 d 2 2 x x y y (x 1, y 1 ) x 2 - x 1

11 Theorem 13-2 (Standard Form for the equation of a Circle) An equation of a circle with center (a, b) and radius r is (x - a) 2 + (y - b) 2 = r 2. (x, y) r y - b (a, b) x - a

12 Sample Problems Find the distance between the points. If necessary, you may draw graphs but you should NOT need to use the distance formula. 1. (- 2, - 3) & (- 2, 4) What should you do first? Plot the points. y What is the distance between these 2 points? (-2, 4) (-2, -3) units x

13 Sample Problems Find the distance between the points. If necessary, you may draw graphs but you should NOT need to use the distance formula. 3. (3, - 4) & (- 1, - 4) What should you do first? Plot the points. y What is the distance between these 2 points? 4 units x (-1, -4) (3, -4)

14 Sample Problems Use the distance formula to find the distance between the 2 points. 5. (- 6, - 2) & (- 7, - 5) What should you do first? (-7, -5) (-6, -2) y P 1 =? P 2 =? P 1 = (- 6, - 2) P 2 = (-7, - 5) x 1 =? y 1 =? x 2 =? y 2 =? x 1 = - 6, y 1 = - 2, x 2 = - 7, y 2 = - 5 What is the distance formula? d x x y y x d d 1 3 d u n its

15 Sample Problems 6 Use the distance formula to find the distance between the 2 points. 7. (- 8, 6) & (0, 0) What should you do first? (-8, 6) 8 10 y (0, 0) P 1 =? P 2 =? P 1 = (0, 0) P 2 = (- 8, 6) x 1 =? y 1 =? x 2 =? y 2 =? x 1 = 0, y 1 = 0, x 2 = - 8, y 2 = 6 What is the distance formula? d x x y y x d d d u n its A triangle with sides 6, 8, and 10. Sound familiar?

16 Sample Problems Find the distance between the points named. Use any method you choose. 9. (5, 4) & (1, - 2) 11. (- 2, 3) & (3, - 2)

17 Sample Problems Given the points A, B & C. Find AB, BC & AC. Are A, B & C collinear? If so, which point is in the middle? 13. A(0, 3) B(- 2, 1) C(3, 6) 15. A(- 5, 6) B(0, 2) C(3, 0)

18 Sample Problems Find the center and the radius of each circle. 17. (x + 3) 2 + y 2 = 49 What is the Standard Form for the equation of a circle? An equation of a circle with center (a, b) and radius r is (x - a) 2 + (y - b) 2 = r 2. (x + 3) 2 + y 2 = 49 (x + 3) 2 + (y 0) 2 = 49 x a = x + 3 a = - 3 y b = y - 0 b = 0 center (a, b) = center (-3, 0) r 2 = 49 radius = 7 units

19 Sample Problems Find the center and the radius of each circle. 19. (x - j) 2 + (y + 14) 2 = 17 What is the Standard Form for the equation of a circle? An equation of a circle with center (a, b) and radius r is (x - a) 2 + (y - b) 2 = r 2. (x - j) 2 + (y + 14) 2 = 17 x a = x - j a = j y b = y + 14 b = -14 center (a, b) = center (j, -14) r 2 = 17 radius = 17 units

20 Sample Problems Write an equation of the circle with the given center and radius. 21. C(3, 0) r = C(- 4, - 7) r = 5 What is the Standard Form for the equation of a circle? An equation of a circle with center (a, b) and radius r is (x - a) 2 + (y - b) 2 = r 2. center (3, 0) = center (a, b) a =? b =? a = 3, b = 0 (x - 3) 2 + (y - 0) 2 = 8 2 (x - 3) 2 + y 2 = 64 center (-4, -7) = center (a, b) a =? b =? a = -4, b = -7 (x (-4)) 2 + (y (-7)) 2 = 5 2 (x + 4) 2 + (y + 7) 2 = 25

21 Section 13-2 Slope of a Line Homework Pages : 1-24 Excluding 10

22 Objectives A. Understand the terms linear equation and slope of a line. B. Understand and identify lines with positive, negative, zero, and undefined slopes. C. Calculate the slopes of various lines. D. Use the slope of a line to graph linear equations.

23 Linear Equation A linear equation is any equation where the graph of the solution set is a line. Example: y = 2x (How many solutions are there to this equation?) Would it be possible to list ALL of the solutions? SOME solutions! If x = Then y = (2x) 0 0 Plot the points. y 2 1 Draw the line. How many points are on a line? x 2 4

24 Linear Equation NOTE! All values of x and y that satisfy the equation y = 2x form a point (x, y) that is ON the line. NOTE! All coordinates (x, y) of points on the line make the equation y = 2x true! Therefore, the GRAPH represents ALL of the solutions to the equation! SOME solutions! If x = Then y = (2x) Plot the points. y x

25 Slopes of Lines slope m rise run y x y x 2 2 y x 1 1 lines with positive slope: rise to the right lines with negative slope: rise to the left steeper line: has a slope with a greater absolute value. the slope of a horizontal line: is zero the slope of a vertical line: is undefined

26 14 (x 2, y 2 ) (13, 14) rise = y = y 2 - y 1 7 (x 1, y 1 ) (8, 7) run = x = x 2 - x 1 y = 14-7 x = 13-8 slope m rise run y x y x 2 y x

27 slope m rise run y x y x 2 2 y x 1 1 slope = 0 slope is undefined

28 Sample Problems Find the slope of the line through the given points. 7. (7, 2) & (2, 7) y P 2 (2, 7) What should you do first? Label the points: P 1 = (7, 2) P 2 = (2, 7) Label the coordinates of the points: x 1 = 7, y 1 = 2, x 2 = 2, y 2 = 7 Write the formula! rise y y y slope m run x x x P 1 (7, 2) x Fill in the blanks! y y x x slope 1 2 1

29 Sample Problems Find the SLOPE and the length of AB 15. A(0, - 9) & B(8, - 3) What should you do first? y Label the points: P 1 = A = (0, -9) P 2 = B = (8, -3) x P 2 (8, -3) slope Label the coordinates of the points: x 1 = 0, y 1 = -9, x 2 = 8, y 2 = -3 Write the formula! rise y y y m run x x x P 1 (0, -9) slope Fill in the blanks! y x y x

30 Sample Problems Find the slope and the LENGTH of AB 15. A(0, - 9) & B(8, - 3) y Label the points: P 1 = A = (0, -9) P 2 = B = (8, -3) Label the coordinates of the points: x 1 = 0, y 1 = -9, x 2 = 8, y 2 = P 1 (0, -9) x P 2 (8, -3) Write the formula! 2 2 distance d x x y y Fill in the blanks! distance d

31 Sample Problems Find one point to the left and to the right of the given point on the same line. 17. P(- 3, 0) slope = 5 2 up 2 y right 5 x Plot the point. If numerator of slope is positive, go up y-units. (if numerator negative, go down y-units) If denominator of slope is positive, go right x-units. (if denominator negative, go left x-units) New point (2, 2)

32 Sample Problems Find one point to the left and to the right of the given point on the same line. 17. P(- 3, 0) slope = 5 2 y up 2 right x ?? If numerator of slope is negative, go down y-units. If denominator of slope is negative, go left x-units. New point (-8, -2)

33 Sample Problems Find the slope of the line through the given points. 3. (1, 2) & (3, 4) 5. (1, 2) & (- 2, 5) 9. (6, - 6) & (- 6, - 6) 11. (- 4, - 3) & (- 6, - 6) Find the slope and the length of AB 13. A(- 3, - 2) & B(7, - 6)

34 Sample Problems Find one point to the left and to the right of the given point on the same line. 17. P(- 3, 0) slope = 19. P(0, - 5) slope = Show point P, Q and R are collinear by showing PQ and QR have the same slope. 21. P(- 8, 6) Q(- 5, 5) R(4, 2) Complete A line with slope passes through points (2, 3) & (10,?) 4

35 Section 13-3 Parallel and Perpendicular Lines Homework Pages : 1-20 Excluding 14

36 Objectives A. Understand the relationship between the geometric and algebraic definitions of parallel and perpendicular lines. B. Determine if 2 lines are parallel, perpendicular, or neither. C. Find the slopes of lines parallel or perpendicular to a given line.

37 Geometric versus Algebraic speak The number 1 lesson from this chapter is to show the tight linkage between geometry and algebra. One key to understanding this linkage is to understand the different dialects we speak in the mathematical language. Relate this to the many dialects within the United States. The vast majority of citizens of the United States speak an English dialect. In other words, the roots of our language are the same whether we are from the deep south or from the industrial northeast. However, the words we use and the way we pronounce them are different, thus creating the dialect. We mean the same thing, but say the things differently.

38 Geometric versus Algebraic speak - continued When we speak geometrically, we use terms such as: Points Lines Planes Angles Polygons Solids When we speak algebraically, we use terms such as: Numbers Variables Equations Sets Inequalities Both the geometric and algebraic dialects are a part of the language of mathematics.

39 Parallel Lines From Chapter 3, the geometric definition of parallel lines is: Two coplanar lines that do not intersect. Theorem 13-3 give the algebraic definition of parallel lines: Two non-vertical lines are parallel if and only if their slopes are equal. In other words, if m 1 =m 2, then line 1 is parallel to line 2.

40 Perpendicular Lines From Chapter 3, the geometric definition of perpendicular lines is: Two coplanar lines that intersect at right angles. Theorem 13-4 give the algebraic definition of perpendicular lines: Two non-vertical lines are perpendicular if and only if the product of their slopes is (- 1). In other words, if (m 1 x m 2 = -1), then line 1 is perpendicular to line 2.

41 Sample Problems 1. Find the slope of (a) AB (b) any line parallel to AB (c) any line perpendicular to AB. A(- 2, 0) & B(4, 4) A (-2, 0) y What should you do first? Label the points: A = P 1 = (-2, 0); B = P 2 = (4, 4) Label the coordinates of the points: x 1 = -2, y 1 = 0, x 2 = 4, y 2 = 4 rise y y2 y1 B (4, 4) slope m run x x2 x1 y2 y slope x x x 2 1 Slope of a parallel line? Two non-vertical lines are parallel if and only if their slopes are equal. 2 slope of parallel line 3

42 Sample Problems 1. Find the slope of (a) AB (b) any line parallel to AB (c) any line perpendicular to AB. A(- 2, 0) & B(4, 4) A (-2, 0) y m What should you do first? Label the points: A = P 1 = (-2, 0); B = P 2 = (4, 4) Label the coordinates of the points: x 1 = -2, y 1 = 0, x 2 = 4, y 2 = 4 rise y y2 y1 B (4, 4) slope m run x x2 x1 y2 y slope x x m2 1 x m 2 1 Slope of a perpendicular line? Two non-vertical lines are perpendicular if and only if the product of their slopes is (- 1). 1 3 slope of perpendicular line 2

43 3. OEFG is a parallelogram. What is the slope of each side? What else do you know? Sample Problems What are the coordinates of point O? O (0, 0) Label the coordinates of the points: x 1 = 0, y 1 = 0, x 2 = 2, y 2 = 7 slope m rise y y2 y1 run x x2 x1 slope EO? slope EO slope FG? O E(2, 7) FG EO so? 7 slope FG 2 G F slope OG? slope OG 0 slope EF? slope EF 0

44 Sample Problems 5. What is the slope of LM & PN? Why are they parallel? What is the slope of MN & LP? Why are they not parallel? What kind of quadrilateral is LMNP? slope LM? slope PN? slope MN? slope LP? slope LM slope PN slope MN slope LP What kind of quadrilateral is LMNP? M(- 4, 2) L(- 3, - 1) LM PN why? MN N(2, 4) P(4, - 2) not parallel to MN why? Trapezoid

45 Sample Problems 7. Find the slope of each side and each altitude of ABC. A(0, 0) B(7, 3) C(2, - 5) slope of altitude from AC =? 2 5 slope of altitude from CB =? 5 8 A (0, 0) y B (7, 3) C (2, -5) What should you do first? slope AB? slope AB slope AC? slope AC slope CB? 35 8 slope CB What must be true of the altitude x drawn from the line containing AB? Altitude must be to the line containing AB. slope AB slope of altitude = -1 3 slope of altitude = slope of altitude from AB = - 3

46 Sample Problems 9. Identify the legs of right RST. R(- 3, - 4) S(2, 2) T(14, - 8) 11. Given parallelogram ABCD, A(- 6, - 4) B(4, 2) C(6, 8) D(- 4, 2). Show that opposite sides are parallel and opposite sides are congruent. 13. R(- 4, 5) S(- 1, 9) T(7, 3) U(4, - 1) Show that RSTU is a rectangle. Show that the diagonals are congruent. Decide what type of quadrilateral HIJK is, then tell why. 15. H(0, 0) I(5, 0) J(7, 9) K(1, 9) 17. H(7, 5) I(8, 3) J(0, - 1) K(- 1, 1) 19. Point N(3, - 4) is on the circle x 2 + y 2 = 25. Find the slope of the line tangent to the circle at N.

47 Section 13-4 Vectors Homework Pages : 1-30 Excluding 8, 16

48 Objectives A. Understand and utilize the terms vector, magnitude, and direction. B. Identify and calculate the horizontal and vertical components of a vector. C. Recognize and identify equal, parallel, and perpendicular vectors. D. Calculate the magnitude and slope of a vector. E. Perform vector addition, vector subtraction, and scalar multiplication.

49 Vectors vector: a line segment with both magnitude and direction, written as an ordered pair of numbers (H, V) where H is the horizontal component and V is the vertical component. y H x V

50 Vector Components horizontal component: the distance traveled left or right from the starting to the ending point, found by taking x stop - x start vertical component: the distance traveled up or down from the starting point to the ending point, found by taking y stop - y start (3, 5) Vector (H, V) = (4, 3) H = Stop Start = 7 3 = 4 (7, 8) V = Stop Start V = 8 5 = 3

51 Vector Terms Equal Vectors vectors with the same magnitude and direction Note that equal vectors may or may not be collinear Magnitude of a Vector The length of the vector. Found by using the distance formula

52 dot product: (H 1, V 1 ) (H 2,V 2 ) = H 1 H 2 + V 1 V 2 vector addition: (H 1, V 1 ) + (H 2,V 2 ) = (H 1 + H 2, V 1 + V 2 ) vector subtraction: (H 1, V 1 ) - (H 2,V 2 ) = (H 1 - H 2, V 1 - V 2 ) scalar multiplication: k(h 1,V 1 ) = (kh 1, kv 1 ) 2 2 magnitude : AB H V V slope: H

53 Find AB and AB 3. A(6, 1) B(4, 3) y B (4, 3) Sample Problems A (6, 1) What should you do first? Order of vector dictates that Point A is the start point and Point B is the stop point. AB? AB ( HV, ) H = x stop x start = 4 6 = -2 V = y stop y start = 3 1 = 2 AB ( HV, ) ( 2,2) x AB? AB H V 2 2 AB 2 2 ( 2) (2) units

54 Sample Problems Perform the scalar multiplication (4, - 1) The scalar k is 3. The horizontal component H is 4. The vertical component V is -1. k(h,v) = (kh, kv) 3(4, -1) = (3 x 4, 3 x -1) = (12, -3)

55 u u Sample Problems Find the vector sum. 21. (3, - 5) + (4, 5) (H 1, V 1 ) + (H 2, V 2 ) =? (H 1, V 1 ) + (H 2, V 2 ) = (H 1 + H 2, V 1 + V 2 ) H, V 3, 5 H, V 4,5,,,,, 3 4, 5 5 7, u u u H V H V u u u H V H V H H V V ? u u u,,, 3 4, 5 5 1, 10 u u u H V H V H H V V

56 Sample Problems Another look at #21! Find the vector sum. 21. (3, - 5) + (4, 5) y AB (3, 5) BC (4,5) AC AB BC AC (3, 5) (4,5) AC (3, 5) (4,5) (3 4, 5 5) (7,0) A(0, 0) C (7, 0) x B (3, -5)

57 Sample Problems 27. An object, K, is being pulled by two forces KX = (- 1, 5) and KY = (7, 3). What single force has the same effect as the two forces acting together? What is the magnitude of this force? KX = (- 1, 5) KY = (7, 3) X(-1, 5) y K(0, 0) Y(6, 8) R(6, 8) The result of 2 forces working on an object at the same time has the SAME result as the object being worked on by the first force and THEN being worked on by the second force. KR KX KY x KR ( 1,5) (7,3) (6,8) KR H V 2 2 KR units

58 Sample Problems Find AB and AB 1. A(1, 1) B(5, 4) 5. A(3, 5) B(- 1, 7) 7. A(0, 0) B(5, - 9) 9. A(- 1, - 1) B(- 4, - 7) Perform the scalar multiplication , (6, - 4)

59 Sample Problems 17. The vectors (8, 6) and (12, k) are parallel. Find k. 19. The vectors (8, k) and (9, 6) are perpendicular. Find k. Find the vector sum. 23. (- 3, - 3) + (7, 7) 25. (7, 2) + 3(- 1, 0)

60 Sample Problems 29. M is the midpoint of AB. T is the trisector of AB. A(2, 3) B(20, 21). Find AB, AM and AT. Find the coordinates of M & T. B(20, 21) T M A(2, 3)

61 Section 13-5 The Midpoint Formula Homework Pages : 1-20 Excluding 18

62 Objectives A. Understand and utilize the midpoint formula.

63 Theorem 13-5 (Midpoint Formula) The midpoint (x m, y m ) of a segment that joins the points (x 1, y 1 ) and (x 2, y 2 ) is the point x y y x, y 1 + x 2, 1 2 m m. 2 2 (x 1, y 1 ) (x m, y m ) (x 2, y 2 )

64 Sample Problems Find the coordinates of the midpoint. 3. (6, - 7) & (- 6, 3) P 2 (-6, 3) y x (0, -2) P 1 (6, -7) midpoint? x x y y midpoint, 2 2 P What should you do first? , 7 P 6,3 1 2 x 6, y 7 x 6, y midpoint, 0, Is this a REASONABLE answer?

65 Sample Problems Find the length, slope and midpoint of PQ. 7. P(3, - 8) Q(- 5, 2) Q (-5, 2) slope? y2 y1 slope x2 x1 28 slope y What should you do first? Label the points: P 1 = P = (3, -8), P 2 = Q = (-5, 2) Label the coordinates of the points: x 1 = 3, y 1 = -8, x 2 = -5, y 2 = x midpoint? P (3, -8) length distance =? length x x y y length units x x y y midpoint, midpoint, 1, 3 2 2

66 Sample Problems 15. Find the midpoints of the legs and then the length of the median of the trapezoid CDEF with vertices C(- 4, - 3), D(- 1, 4), E(4, 4) & F(7, - 3). What should you do first? y D(-1,4) ,, xc xd yc yd midpoint CD? midpoint CD, midpoint CD,, E(4,4) midpoint EF? x mid EF mid EF x x y y, 2 2 E F E F ,, C(-4,-3) F(7,-3)

67 Sample Problems 15. Find the midpoints of the legs and then the length of the median of the trapezoid CDEF with vertices C(- 4, - 3), D(- 1, 4), E(4, 4) & F(7, - 3). y length distance =? 2 2 distance x x y y D(-1,4) E(4,4) ,, x distance C(-4,-3) F(7,-3)

68 Sample Problems Find the coordinates of the midpoint. 1. (0, 2) & (6, 4) 5. (2.3, 3.7) & (1.5, - 2.9) Find the length, slope and midpoint of PQ. 9. P(- 7, 11) Q(1, - 4) M is the midpoint of AB. Find the coordinates of B. 11. A(1, - 3) M(5, 1) 13. A(0, 0) & B(8, 4), show P(2, 6) is on the perpendicular bisector of AB.

69 Sample Problems 17. Show that OQ & PR have the same midpoint. What kind of quadrilateral is OPQR? Show that the opposite sides are parallel. Show the opposite sides are congruent. P(2, 6) Q(9, 9) R(7, 3) O

70 Sample Problems 19. In right OAT M is the midpoint of AT. What are the coordinates of M? Find MA, MT & MO. Find the equation of the circle that circumscribes the triangle. A(0, 8) M T(- 6, 0) O

71 Section 13-6 Graphing Linear Equations Homework Pages : 1-33 ALL Excluding 1, 3, 32

72 Objectives A. Identify linear equations. B. Understand and utilize the standard form of a linear equation. C. Understand and utilize the slopeintercept form of a linear equation. D. Understand and apply multiple methods for solving systems of linear equations.

73 Remember! A linear equation is any equation whose solution graph is a line.

74 Methods for Graphing a Line: 1. Plot two or more points Remember any two points are contained on one unique line. For graphing purposes, I recommend you plot at least 3 points y 2x 1 y If x =? Then y = x

75 Theorem 13-6 (Standard Form of a Linear Equation) The graph of any equation that can be written in form Ax + By = C where A is zero or positive, A and B are not both zero, and A, B and C are integers, is a line. Therefore, Ax + By = C is the standard form of a linear equation. A slope B x - intercept is y - intercept is the the point point C A,0 C 0, B

76 Theorem 13-7 (Slope Intercept Form of a Line) A line with the equation y = mx + b has slope m and y-intercept b.

77 Methods for Graphing a Line: 1. Plot two points 2. Plot one point (Y-intercept) and rise and run according to the slope. y 2x 1 y = mx + b Slope-intercept form of a linear equation is? y m 2 or m b = y-intercept = (0, 1) x

78 Solving Systems of Linear Equations - Graphing Graph the two linear equations and identify the intersection point. The point of intersection of the two lines gives the x and y coordinates that will make BOTH linear equations true. Can be used on any system but your answer is only as accurate as your graph.

79 Solving Systems of Linear Equations - Graphing y y x 1 x 1 y Point of intersection (-1, 0) CHECK your answer! y x 1 ( 1) 1 0 y x 1 ( 1) x

80 Solving Systems of Linear Equations Isolate and Substitute Solve for (isolate) x or y in one equation. Substitute the expression from the step above into the second equation. Solve the second equation for the remaining variable. Substitute the answer from the step above into either original equation and solve for the other variable. Can be used on any system but works best when one coefficient is either 1 or - 1.

81 Solving Systems of Linear Equations Isolate and Substitute 2x y 12 3x 2y 17 Solution (1, 10) Isolate (solve for) y in the first equation y = 12 2x Using the result of the isolation, substitute for y in the second equation 3x 2(12 2x) = -17. Solve for x 3x x = -17 7x = 7 x = 1 Substitute x value back into original equation 2(1) + y = 12 y = 10

82 Solving Systems of Linear Equations Linear Combination Addition: add the two equations to eliminate one of the variables. Works only if the coefficients of one of the variables are opposites. Subtraction: subtract the two equations to eliminate one of the variables. Works only if the coefficients of one of the variables are the same. Multiply one or both equations by a constant in order to create coefficients of the same variable that are either the same or opposites, then add/subtract.

83 5x 2y 3x 4y 16 Solving Systems of Linear Equations Linear Combination 6 5(2) 2y 16 Solution (2, 3) 2y 6 What to do? y 3 Graphing would be complicated. Isolating x or y would leave nasty fractions. Adding or subtracting equations will not eliminate variable. Multiply both sides of first equation by 2 10x + 4y = 32 Now add the two equations: 10x 4y 3x 4y 13x 0y x

84 Sample Problems Find the x & y intercepts and the slope, then graph. 7. 3x + y = - 21 How do you find the x-intercepts for a solution graph? 3x + 0 = -21 x = -7 Plug in zero for the y-value and solve for the x-value. x-intercept = (-7, 0) How do you find the y-intercepts for a solution graph? 3(0) + y = -21 Plug in zero for the x-value and solve for the y-value. y = -21 y-intercept = (0, -21) How do you find the slope of the solution graph? The equation 3x + y = -21 is in which form? m Ax + By = C A 3 m B 1 3 y = -3x - 21 The equation 3x + y = -21 is in standard form. What is the standard form of a linear equation? where A is positive or zero, A and B are not both zero, and A, B, and C are integers. Slope of the line when using standard form =? Any other way of finding the slope? Convert the equation into slope-intercept form!

85 Sample Problems Find the x & y intercepts and the slope, then graph. 7. 3x + y = - 21 x-intercept = (-7, 0) y-intercept = (0, -21) 3 m 1 Graph the solution. (-7, 0) y x (0, -21)

86 Solve the system 29. 4x + 5y = - 7 2x - 3y = 13 Sample Problems Multiply this equation by 2 4x 6y = 26 What are the 3 methods you can use to solve systems of linear equations? 1. Graph 2. Isolate and substitute 3. Linear combination Which of these would you use? Why would you NOT use the graphing method for this problem? Linear combination will be the method you use most frequently! What would you do in order to do linear combination to this system? 4x + 5y = (4x - 6y = 26) x + 11y = -33 y = -3 Plug back into equation! Solution (2, -3) 4x + 5(-3) = - 7 4x 15 = -7 4x = 8 x = 2

87 1. Graph y = mx if m = 2, - 2, Sample Problems 3. Graph 1 y x b for b = 0, 2, - 2, Graph y = 0, y = 3, y = , Find the x & y intercepts and the slope, then graph. 9. 3x + 2y = x + 8y = 20

88 Sample Problems Find the slope, x & y intercepts, then graph. 13. y = 2x y = - 4x 17. y 2 3 x x + y = x - 2y = x - 4y = 6

89 Sample Problems Solve the system 25. x + y = 3 x - y = x + 2y = 10 3x - 2y = 6

90 Section 13-7 Writing Linear Equations Homework Page 555: 1-28 Excluding 6, 16 Final Answers must be in STANDARD FORM for a linear equation (Ax + By = C)

91 Objectives A. Understand and utilize the pointslope form of a linear equation. B. Use various pieces of information about a line or linear equation to determine the standard, slopeintercept, and/or point-slope form of the linear equation.

92 Theorem 13-8 (Point Slope Form of a Line) An equation of the line that passes through the point (x 1, y 1 ) and has slope m is y - y 1 = m(x - x 1 ).

93 Writing an Equation of the Line Find the slope and one point on the line. Use the point-slope form of a line; put the slope in for m and the point in for (x 1, y 1 ). Distribute and rearrange the equation until it is in standard form, with all coefficients being integers.

94 Sample Problems Write the equation of the line in standard form slope = y-intercept (0, 8) 5 What is the point-slope form of a linear equation: y y mx x y8 x x y Fill in the given point and slope: y8 x 0 Put in standard form (Ax + By = C): 7x5y40

95 Sample Problems 25. Write the standard form of the equation of a line through (5, 7) and parallel to the line y = 3x 4. y = 3x 4 is in what form? What is the slope-intercept form of a linear equation? y = mx + b where m is the slope and b is the y-intercept. What is be the slope of the line y = 3x - 4? m 1 = 3 What will be the slope of the new line? If the lines are to be parallel, then, m 1 = m 2. So m 2 = 3. You are given the point (5, 7). What is the point-slope formula? 1 1 Fill in what you know. Put in standard form. y y m x x y7 3 x5 y 7 3x15 3x y 8

96 Sample Problems Write the equation of the line in standard form. 1. slope = 2 y-intercept = (0, 5) 1 3. slope = y-intercept (0, - 8) 2 5. slope = 7 y-intercept (0, 8) 5 7. x-intercept (8, 0) y-intercept (0, 2) 9. x-intercept (- 8, 0) y-intercept (0, 4) 11. point (1, 2) slope = point (- 3, 5) slope = 1 3

97 Sample Problems 15. point (- 4, 0) slope = line through (1, 1) & (4, 7) 19. line through (- 3, 1) & (3, 3) 21. vertical line through (2, - 5) 23. line through (5, - 3) and parallel to the line x = line through (- 3, - 2) and perpendicular to the line 8x - 5y = perpendicular bisector of the segment joining (0, 0) & (10, 6) 31. the line through (5, 5) that makes a 45 angle measured counterclockwise from the positive x axis

98 Section 13-8 Organizing Coordinate Proofs Homework Pages : 1-10 Draw and label each diagram on graph paper!

99 Objectives A. Understand and apply the term coordinate proof. B. Given a polygon, choose an appropriate placement of coordinate axes on the polygon to assist in a coordinate proof. C. Given a polygon properly placed on a coordinate plane, choose appropriate coordinates (variable and fixed) of the vertices of the polygon to assist in a coordinate proof.

100 Coordinate Proofs A coordinate proof: Is a method of proving a conditional statement Is similar to the direct and indirect proof methods you have already encountered Has all of the requirements of any other proof method Identify the conditional statement to be proved Identify the given information and diagrams Identify the proof statement Provide logical steps from given information to proof statement Can be used in conjunction with a direct or indirect proof. Uses objects placed in a coordinate plane to assist with the proof Uses variable coordinates from the coordinate plane to provide a GENERAL proof of the conditional statement.

101 Coordinate Proof: Placing the Object on the Coordinate Plane Consider placing an isosceles right triangle on a coordinate plane. What are the characteristics of an isosceles right triangle? Polygon with 3 sides Has exactly one right angle Has congruent legs

102 Coordinate Proof: Placing the Object on the Coordinate Plane Placing an isosceles right triangle on a coordinate plane. y Does this appear to be a good placement of the triangle in the coordinate plane? Why or why not? x Is it easy to prove there is a right angle in the triangle? Is it easy to prove there are congruent legs in the triangle? Are the coordinates of the vertices of this triangle fixed real numbers? If we wanted to generalize the proof, how many variable coordinate would be required? Would calculation of the slopes of the lines be an easy task? Would calculation of the lengths of the sides be an easy task?

103 Coordinate Proof: Placing the Object on the Coordinate Plane Placing an isosceles right triangle on a coordinate plane. y Can we make our life easier? Is there any rule that disallows us from rotating the triangle on the coordinate plane? NO! So choose wisely! x 1 2 3

104 Coordinate Proof: Placing the Object on the Coordinate Plane Placing an isosceles right triangle on a coordinate plane. y (0,?) a) x Is it easy to prove there is a right angle in the triangle? What is the x-coordinate of this point? What is the y-coordinate of this point? Be sure that it is useful in a general proof! What is the y-coordinate of this point? What is the x-coordinate of this point? Be sure that it is useful in a general proof and it makes an isosceles triangle! What is the x-coordinate of this point? (0,?) 0) (?, (a, 0) What is the y-coordinate of this point?

105 Coordinate Proof: Placing the Object on the Coordinate Plane Placing an isosceles right triangle on a coordinate plane. y (0, a) x Is it easy to prove there are congruent legs in the triangle? Are the coordinates of the vertices of this triangle fixed real numbers? How many variable coordinate would be required? Would calculation of the slopes of the lines be an easy task? Would calculation of the lengths of the sides be an easy task? (0, 0) (a, 0)

106 Coordinate Proof: Placing the Object on the Coordinate Plane Steps to properly placing a figure on a coordinate plane to assist in a coordinate proof: 1. If one or more right angles exist in the figure, place one of them at the intersection of the coordinate axes (origin). 2. If one or more sets of parallel lines exist in the figure, place at least one of the parallel sides on either the x-axis or the y-axis. 3. In MOST figures, it is best to use the origin as one of the vertices of the figure. 4. Whenever possible, place other vertices on the x-axis (x, 0) or on the y-axis (0, y).

107 Organizing Coordinate Proofs Some Sample Diagrams (0, (?, b)?) scalene right triangle (?, (b,?) c) scalene triangle (?, (0, 0)?) (?, (a,?) 0) (?, (0,?) 0) (a, (?, 0)?) (?, (0,?) b) scalene triangle (?, (a,?) 0) (?, (c,?) 0)

108 Organizing Coordinate Proofs isosceles triangle (?, (0,?) b) isosceles triangle (?, (a,?) b) (?, (-a,?) 0) (?, (a,?) 0) (0, (?, 0)?) (2a, (?,?) 0) (?, (b,?) c) (a(?, + b,?) c) (?, (0,?) a) (?, (b,?) a) (?, (0,?) 0) rectangle (?, (b,?) 0) (0, (?, 0)?) (?, (a,?) 0) parallelogram

109 Organizing Coordinate Proofs (?, (b,?) c) (?, (a -?) b, c) (?, (b,?) c) (?, (d,?) c) (0, (?, 0)?) (?, (a,?) 0) isosceles trapezoid (0, (?, 0)?) (?, (a,?) 0) trapezoid

110 Sample Problems Supply the missing coordinates without using any new variables. (?,?) (?,?) (?,?) (a, b) (m, n) 1. rectangle (?,?) (?,?) (- f, 0) (f, 0) 3. square (?,?) 5. parallelogram (h, 0) (s, 0) 7. equilateral triangle

111 Sample Problems (?, b) (?,?) 9. rhombus (c, 0)

112 Section 13-9 Coordinate Geometry Proofs Homework Page 562: 1-10 Make sure you do two-column proofs and include a properly annotated diagram!

113 Objectives A. Prove conditional statements by using coordinate geometry proof methods.

114 Writing Coordinate Proofs To write a coordinate proof of a geometric theorem the first step is to place the diagram on a generic graph i.e. a graph without numbers. The second step is to label each vertex with variable coordinates. Each x-coordinate and each y-coordinate must be assigned a different variable unless a relationship has already been proven to exist. To minimize the number of letters used to label the vertices, it is wise to place one vertex on the origin and as many other vertices as possible on the coordinate axes.

115 Writing Coordinate Proofs The third step is to write the given and the proof statements as algebraic expressions, using the coordinates from the diagram. The fourth step is to create the body of the proof by using the rules of algebra to transform the given equation into the equation in the prove statement. Write as two-column proofs. Most proofs can be accomplished by using the equations from theorems 13-1 to 13-7 along with the definitions of slope and vectors to write or transform your given statement.

116 Coordinate Geometry Proofs - Example Prove that the diagonals of an isosceles trapezoid are congruent. First step? y R (0, 0) U (n, 0) Diagram on a generic graph (without numbers). Second step? S (p, q) T (n - p, q) The second step is to label each vertex with variable coordinates. Each x-coordinate and each y-coordinate must be assigned a different variable unless a relationship has already been proven to exist. x Third step? The third step is to write the given and the proof statements as algebraic expressions, using the coordinates from the diagram. How would you show these diagonals to be congruent? Distance formula? d x x y y RT n p q n p q US n p q n p q Since RT = US, the diagonals are congruent.

117 Coordinate Geometry Proofs - Example Prove that the diagonals of an isosceles trapezoid are congruent. Fourth step? y S (p, q) T (n - p, q) The fourth step is to create the body of the proof by using the rules of algebra to transform the given equation into the equation in the prove statement. R (0, 0) U (n, 0) x 1. Quadrilateral RSTU; ST RU; RS TU; diagram with coordinates given d x x y y Given 2. The distance d between points (x, y ) and (x, y ) is given by d x x y y

118 Coordinate Geometry Proofs - Example Prove that the diagonals of an isosceles trapezoid are congruent. Fourth step? y S (p, q) T (n - p, q) The fourth step is to create the body of the proof by using the rules of algebra to transform the given equation into the equation in the prove statement. R (0, 0) U (n, 0) x RT n p 0 q 0 n p q US n p 0 q n p q 3. Substitution and simple math. 4. Substitution and simple math. 5.US RT 5. Substitution. 6.US RT 6. Definition of congruence

119 Sample Problems Prove that the segment joining the midpoints of the legs of a trapezoid is parallel to the bases and has a length equal to half the sum of the lengths of the bases. y E N (b, c) M (d, c) F O P (a, 0) What are the coordinates of point E? x What might be useful to add to the diagram? What is the midpoint formula? b 0 c 0 b c E midpoint,, ON What are the coordinates of point F? F x x y y midpoint, 2 2 d a c 0 d a c midpoint,, MP

120 m Sample Problems Prove that the segment joining the midpoints of the legs of a trapezoid is parallel to the bases and has a length equal to half the sum of the lengths of the bases. y E N (b, c) M (d, c) O P (a, 0) EF mop?? m EF E b c, 2 2 F d a c, 2 2 F How would you prove EF OP MN? What is the slope formula? y2 y1 m x x 2 x 1 c c d a b d a b m? MN c c 0 0 m 0 MN a 0 a d b d b m OP

121 Sample Problems Use coordinate geometry to prove each statement. First draw a figure and choose convenient axes and coordinates. 1. The diagonals of a rectangle are congruent. (Theorem 5-12)

122 Sample Problems Use coordinate geometry to prove each statement. First draw a figure and choose convenient axes and coordinates. 3. The diagonals of a rhombus are perpendicular. (Theorem 5-13) (Hint: Let the vertices be (0, 0), (a, 0), (a + b, c), and (b, c). Show that c 2 = a 2 b 2.)

123 Sample Problems 5. Prove that the segment joining the midpoints of the diagonals of a trapezoid is parallel to the bases and has a length equal to half the difference of the lengths of the bases. y N (b, c) M (d, c) O P (a, 0) x

124 Sample Problems 7. Prove that the figure formed by joining, in order, the midpoints of the sides of quadrilateral ROST is a parallelogram. y R (2b, 2c) T (2d, 2e) O S (2a, 0) x

125 Sample Problems 9. Prove that the angle inscribed in a semicircle is a right angle. (Hint: The coordinates of C must satisfy the equation of a circle.)

126 Chapter Thirteen Coordinate Geometry Review Homework Page 568: 1-18