1 UNIT SIX MODERN APPLICATIONS OF PYTHAGORAS S THEOREM Coordinate Systems 124 Distance Formula 127 Midpoint Formula 131 SUMMARY 134 Exercises 135
2 UNIT SIX: 124 COORDINATE GEOMETRY Geometry, as presented by Euclid, stood on its own for almost two millennia. But in the 1600s western European scholars discovered a way to unite the subject with the power of algebra through the newly discovered notion of a coordinate system. One draws two number lines set at 90 o to one another and uses these axes to associate to each point in the plane a pair of numbers. It is conventional to call the horizontal axis, with positive numbers to the right, the x-axis, and the vertical axis, with the positive numbers upwards, the y-axis. Each point ( x, y) is listed with the x-coordinate first and the y-coordinate second.
3 UNIT SIX: 125 SOME JARGON: The point (0,0) is called the origin. The two axes divide the plane into four quadrants. Comment: See the unit on trigonometry for an explanation as to why mathematicians place the quadrants in this counter-clockwise order. HISTORICAL NOTE Many people say that French philosopher and mathematician Renè Descartes ( ) invented the concept of a coordinate system. This is not quite true: Descartes never thought to explicitly draw a vertical axis. He did, however, draw pictures of graphs that seemed to use a second axis paired with a horizontal number line, but he never held that second axis at a fixed 90 o to the horizontal: the angle of the second axis varied from problem to problem. Descartes, however, did manage to solve a large number of challenging questions using coordinate systems. For this reason, he is dubbed the father of coordinate geometry. (And the coordinate system we use is sometimes called a Cartesian coordinate system to honor his name.) No one knows when or how the idea of using a fixed vertical axis came into place. It seems to be an idea that simply evolved during the 1700s.
4 UNIT SIX: 126 Another Comment: There really is nothing special about the way we place the x- and y-axes. We could, if we so decided, use axes as follows at an angle of 83 o oriented as shown: It is a convention that we use vertical and horizontal axes with positive numbers to the right and to the top. It is also a convention that we choose to read the coordinates of a point (2,3) with the first number 2 the x-coordinate and the second number 3 the y-coordinate. However, it turns out to be exceptionally convenient to work with axes that lie at 90 o to one another. As we shall next see, this allows us to make use of Pythagoras s theorem in an easy and natural way.
5 UNIT SIX: 127 THE DISTANCE FORMULA: Here are two points P and Q on the coordinate plane. What is the distance between them? The advantage to using a pair of horizontal and vertical axis is that we are able to draw right triangles at different locations with ease. In this example, we can see that the line segment PQ is the hypotenuse of a right triangles with legs of lengths 6 1= 5 and 5 2= 3. Thus, by Pythagoras theorem: ( ) ( ) ( ) ( ) distance PQ= difference in x-values + difference in y-values = = = 34
6 UNIT SIX: 128 Question: Does the formula: ( ) ( ) distance = difference in x-values + difference in y-values always work? What if the coordinates of some points are negative? Let s examine a few examples to see that this formula is indeed valid in all situations. EXAMPLE: distance = ( ) 3 ( 2) + (4 2) = = 29
7 UNIT SIX: 129 EXAMPLE: Distance = = 68 In general, given two points P= ( x1, y1) and Q= ( x2, y2), one can always draw a right triangle with hypotenuse PQ. The horizontal leg of the triangle has length given by the difference of the x- coordinates (and this is true even with negative values present) and the vertical leg of the triangle has length given by the difference of the y-coordinates. Thus by Pythagoras s theorem we have: DISTANCE FORMULA: The distance between two points P= ( x1, y1) and Q= ( x, y ) is given by: ( ) ( ) PQ = difference in x-values + difference in y-values Comment: This formula can be written: PQ= ( x x ) + ( y y )
8 UNIT SIX: 130 Note: The distance formula has the advantage that it does not matter in which order one decides to subtract the two x-coordinates or the two y-coordinates. e.g. The distance between P= (2,3) and Q= (5,8) can be computed as: PQ= (5 2) + (8 3) = = 34 or as PQ= (2 5) + (3 8) = ( 3) + ( 5) = 34 Reason: Even though x2 x1 and x1 x2 are not the same (in fact, one is the negative of the other), their squares are the same. ( x x ) = ( x x ) PRACTICE EXERCISE: Find the distances between the following pairs of points: a) A= (2,10) B= (2,8) (Does the answer surprise you?) b) C= ( 7,4) D= ( 3,1) c) P= ( 3,3) Q= (8, 2) PRACTICE EXERCISE: Let A= (3, 4), B= (5,10) and R= (4,7). a) Show that AR=RB b) Suppose S = (1, k) is another point with AS = SB. What is the value of k?
9 UNIT SIX: 131 MIDPOINT FORMULA: Suppose we wish to find the coordinates of the midpoint M of a line segment PQ. One might guess that M has x-coordinate given by the value exactly half way x + x between x 1 and x 2, namely, ; and that that y-coordinate of M might be the y1+ y2 value exactly half way between y1 and y 2, namely,. This gives two right 2 triangles with bases of equal lengths and heights of equal length. By Pythagoras s theorem it seems that, in the diagram below, we would indeed have a equal to b making M the midpoint of PQ. But there is a problem. How do we know for sure that this point M with coordinates given as the average of the x values and the average of the y values actually lies on the line PQ? As we have seen, that PM = MQ doesn t automatically imply that M is the midpoint of PQ! It turns out that all is okay and that M given by this approach is indeed the midpoint of the line segment. Proving this, however, is quite tricky. We leave it as an optional exercise for the brave. (Though, in unit 16, we ll see a quick geometric way to prove the formula true.)
10 UNIT SIX: 132 MIDPOINT FORMULA: The midpoint of the line segment connecting P= ( x1, y1) and Q= ( x2, y2) is M x + x y + y =, EXAMPLE: Find the midpoint M of the line segment connecting A= ( 1,3) to B= (2,8). Show thatam = MB to verify that M is indeed equidistant from A and B. Also verify that AM + MB equals AB to show that M does indeed lie on AB. Answer: The midpoint is given by: M =, =,. The distance between 1 11, and ( 1,3) is: AM = = + = + = = The distance between 1 11, and (2,8) is: MB= = + = = These distances are indeed the same. Also, AB= = 34 and so AM + MB does equal AB.
11 UNIT SIX: 133 PRACTICE EXERCISE: a) Suppose A = (3, 4) and T = (6,11). Find the coordinates of a point B so that T is the midpoint of AB. b) Suppose P= (2 k,3 m) and R= (5 k, m). Find the coordinates of a point S so that R is the midpoint of PS.
12 UNIT SIX: 134 SUMMARY OF KEY RESULTS DISTANCE FORMULA: The distance between two points P= ( x1, y1) and Q= ( x, y ) is given by: ( ) ( ) PQ = difference in x-values + difference in y-values ADVICE: Don t memorize this formula! If you realize that this is just Pythagoras s theorem you can easily reconstruct it when you need it. MIDPOINT FORMULA: The midpoint of the line segment connectingp= ( x1, y1) and Q= ( x2, y2) is: x 1+ x 2 1 2, y + M y = ADVICE: Don t memorize this formula! Just realize that the midpoint has coordinates that are the average of the x and y coordinates of the original points.
13 UNIT SIX: 135 EXERCISES: Question 1: Identify the quadrant or other specific location of: a) a point with negative x-coordinate and positive y-coordinate b) a point with y-coordinate zero c) a point with both coordinates negative Question 2: What is the name of the point where the coordinate axes intersect? Question 3: Find the distances between the following pairs of points. (Leave your answers in terms of square roots.) a) (11,0) and (5,9) b) (4, 7) and (7,4) c) (0, 0) and (-2, -5) d) (4, 400) and (-3, 424) e) (-2, 11) and (6, 15) Question 4: Find the coordinates of the midpoint of PQ if a) P = (5,6) and Q = (11, 4) b) P = (-3, 17) and Q = (15, 37) c) P = (-12, 6) and Q = (0, 13) d) P = ( a, 0) and Q = (-a, b) Question 5: a) A= ( a,7) and M = (2 a, 2) and M is the midpoint of AB, what are the coordinates of B? b) Find a value for x so that M = (2x+ 2, 9) is the midpoint of PQ with P= ( x, x+ 3) and Q= (3x+ 4, 2 x). Question 6: A triangle ABC has vertices A= ( 1, 2), B= (1,6) and C= (3,0). Find the coordinates of the midpoint M of BC and the length of the line segment MA. (A line connecting one vertex of a triangle to the midpoint of the opposite side is called a median of the triangle.)
14 UNIT SIX: 136 Question 7: A point P is said to be equidistant from two points A and B ifap= PB. Let A = (-2, 3) and B = (2, -1). a) Show that the point P = (4,5) is equidistant from A and B. b) Is the point Q = (-4, -5) equidistant from A and B? c) Is the point R = (-10, -9) equidistant from A and B? d) CHALLENGE: Show that every point of the form ( k, k+ 1) is equidistant from A and B. Question 8: Find a point with y-coordinate 7 that is 5 units away from (2,3). COMMENT: There are two possible answers. Can you find them both? Question 9: a) Find an equation that must be true for the numbers x and y if the point A= ( x, y) is a distance of 6 units from B= (2,9). b) When Jinny answered this question she came up with the formula: ( x ) ( y ) = 36. Is this correct? Question 10: Let F = (3,4) and G = (13, -6). a) Find the coordinates of the point half-way along FG. (That is, find the midpoint of FG.) b) Find the coordinates of the points that are one-quarter and three-quarters along FG. Question 11: A triangle is said to be isosceles if at least two of its sides are the same length. Let A = (0,2); B = (12, 10) ; C = (4, -2). a) Use the distance formula to show that the triangle formed by the points A, B and C is isosceles. b) Compute the midpoints of the three sides of this triangle. c) Is the triangle formed by the midpoints also isosceles? Question 12: Is there a point with the same x- and y- coordinates that is a distance 10 from the point (3,5)? If so, what is it? (And is there more than one point with this property?)
15 UNIT SIX: 137 Question 13: (OPTIONAL) The goal of this exercise is to prove the midpoint formula: MIDPOINT FORMULA: The midpoint of the line segment connecting P= ( x1, y1) and Q= ( x2, y2) is M x + x y + y =, Warning: The algebra is hefty! First we need to show that PM and MQ have the same value. This requires using the distance formula. First note that: x1+ x x = x + x x = x x x + x x = x x x = x x 2 with similar equations for the y-values a) Show that PM = x x + y y b) Show that MQ is the same! Second we need to show that M is actually on the line segment PQ. We ll do this by showing that PM + MQ= PQ, which requires more algebra!
16 UNIT SIX: 138 We have: PM = x x + y y c) Show that this can be rewritten as: 1 1 PM = ( x x ) + ( y y ) PM = x2 x1 + ( y2 y1 ) 2 d) Show that this can be rewritten as: ( ) Notice that this says that PM is half the value of PQ (as we would have hoped!) We know that MQ has the same value of PM, so MQ is also half of PQ. e) Explain now why PM + MQ= PQ must indeed be true. We re done!
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