2.4 Coordinate Proof Using Distance with Quadrilaterals

Transcription

1 Name Class Date.4 Coordinate Proof Using Distance with Quadrilaterals Essential Question: How can ou use slope and the distance formula in coordinate proofs? Resource Locker Eplore Positioning a Quadrilateral on the Coordinate Plane You have used coordinate geometr to find the midpoint of a line segment and to find the distance between two points. Coordinate geometr can also be used to prove conjectures. Remember that ou previousl learned several strategies that make using a coordinate proof simpler. The are: Use the origin as a verte, keeping the figure in Quadrant I. Center the figure at the origin. Center a side of the figure at the origin. Use one or both aes as sides of the figure. Position a rectangle with a length of 8 units and a width of 3 units in the coordinate plane as described. Method 1 Center the longer side of the rectangle at the origin. (-4, 3) (-4, 0) -4-4 (4, 3) (4, 0) 0 4 B Method Use the origin as a verte of the rectangle. Depending on what ou are using the figure to prove, one method ma be better than the other. For eample, if ou need to find the midpoint of the longer side, use the first method. coordinate proof can also be used to prove that a certain relationship is alwas true. You can prove that a statement is true for all right triangles without knowing the side lengths. To do this, assign variables as the coordinates of the vertices. Position a square, with side lengths a, on a coordinate plane and give the coordinates of each verte. C Sketch the square. Label the side lengths. 4 (0, 3) (8, 3) (0, 0) (8, 0) Module 97 Lesson 4

2 What are the coordinates of each verte? Reflect 1. Discussion Describe another wa ou could have positioned the square and give the coordinates of its vertices.. When writing a coordinate proof wh are variables used instead of numbers as coordinates for the vertices of a figure? Eplain 1 Proving Properties of a Parallelogram You have alread used the Distance Formula and the Midpoint Formula in coordinate proofs. s ou will see, slope is useful in coordinate proofs whenever ou need to show that lines are parallel or perpendicular. Eample 1 Prove or disprove that the quadrilateral determined b the points (4, 4), B (3, 1), C (, 1), and D ( 1, ) is a parallelogram. D 4 Use slopes to write the coordinate proof. To determine whether BCD is a parallelogram, find the slope of each side of the quadrilateral. Slope of B _ = 3; Slope of BC _ ; Slope of CD _ (-1) (-) = 3_ 1 = 3; Slope of D _ (-1) 5 Compare slopes. The slopes of opposite sides are equal. This means opposite sides are parallel. So, quadrilateral BCD is a parallelogram. -4 C B 0 4 Module 98 Lesson 4

3 B Use the Distance Formula to write the coordinate proof. To determine whether BCD is a parallelogram, find the length of each side of the quadrilateral. Remember that the Distance Formula is length = ( - 1 ) + ( - 1 ). B = ( - 4) + (1-4) = (-1) + ( ) = _ CD = (- 1- ) + ( - (-1) ) = (1) + ( ) = _ BC = (- - ) + ( - 1) = (-5) + ( ) = _ D = (4 - ) + (4 - ) = + ( ) ( ) = _ Compare the side lengths. The lengths of the opposite sides are. B the, we can conclude that BCD is a. Reflect 3. Suppose ou want to prove that a general parallelogram WXYZ has diagonals that bisect each other. Wh is it convenient to use general verte coefficients, such as a and b? Your Turn Write a coordinate proof given quadrilateral BCD with vertices (3, ), B (8, ), C (5, 0), and D (0, 0). 4. Prove that BCD is a parallelogram. 5. Prove that the diagonals of BCD bisect each other. Module 99 Lesson 4

4 Eplain Proving Conditions for Special Parallelograms Eample B Prove or disprove each statement about the quadrilateral determined b the points Q (, -3), R (-4, 0), S (-, 4), and T (4, 1). The diagonals of QRST are congruent. The length of SQ _ = The length of RT _ = 65. (-4-4) + (0-1) = 65. ( - (-) ) + (-3-4) = So, the diagonals of QRST are congruent. QRST is a rectangle. Find the slope of each side of the quadrilateral. Slope of QR _ (-3) = 3_ -6 = - 1_ ; Slope of RS _ _ = ; 1 - Slope of ST _ = - - _ 1 - _ 1 - = ; Slope of TQ _ _ = 1 - R -4 S 4 T Q Find the products of the slopes of adjacent sides. (slope of _ QR ) (slope of _ RS ) = = ; (slope of _ RS ) (slope of _ ST ) = = ; (slope of _ ST ) (slope of _ TQ ) = = ; (slope of _ TQ ) (slope of _ QR ) = = You can conclude that adjacent sides are. So, quadrilateral QRST is a. Reflect 6. Eplain how to prove that QRST is not a square. Module 100 Lesson 4

5 Your Turn Prove or disprove each statement about quadrilateral WXYZ determined b the points W (0, 0), X (4, 3), Y (9, 3), and Z (5, 0). 4 W Z X Y 7. WXYZ is a rhombus. 8. The diagonals of WXYZ are perpendicular. Eplain 3 Identifing Figures on the Coordinate Plane Eample 3 Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that appl. (0, ), B (3, 6), C (8, 6), D (5, ) Step 1 Graph BCD. Step Determine if BCD is a rectangle. C = (8-0) + (6 - ) = 80 = 4 5 BD = (5-3) + ( - 6) = 0 = 5 Since 4 5 5, BCD is not a rectangle. Thus, BCD is not a square. Step 3 Determine if BCD is a rhombus. Slope of _ C = 1_ Slope of _ BD = - Since ( 1_ ) (-) = -1, C BD. BCD is a rhombus. 6 B C 4 D Module 101 Lesson 4

6 B E( 4, 1), F( 3, ), G(3, 0), H(, 3) Step 1 Graph EFGH. Step Determine if EFGH is a rectangle. EG = ( 3 - ) + ( 0 - ) FH = + ( - ) ( - (-3) ) = = = = 5 E F H G Since = 5 _, the diagonals are. EFGH a rectangle. Step 3 Determine if EFGH is a rhombus. 0 - (-1) Slope of = _ 3 - (-4) = 1_ 7 ; Slope of = _ (- 3) -5 5 = - 1 Since ( 1_ 7 ) (-1) -1, _ EG is to _ FH. So, EFGH is not a rhombus and cannot be a. Your Turn Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that appl. 9. K( 5, 1), L(, 4), M(3, 1), N(0, 4) 10. P (-4, 6), Q (, 5), R (3, -1), S (-3, 0) Elaborate 11. How can ou use slopes to show that two line segments are parallel? Perpendicular? 1. When ou use the Distance Formula, ou find the square root of a value. When finding the square root of a value, ou must consider both the positive and negative outcomes. Eplain wh the negative outcome is not used in the coordinate proofs in the lesson. Module 10 Lesson 4

7 13. Essential Question Check-In How can ou use slope in coordinate proofs? Evaluate: Homework and Practice 1. Suppose ou have a right triangle. If ou want to write a proof about the midpoints of the legs of the triangle, which placement of the triangle would be most helpful? Eplain.. Use the origin as a verte, keeping the figure in Quadrant I with vertices (0, b), (a, 0), and (0, 0). B. Center the triangle at the origin. C. Use the origin as a verte, keeping the figure in Quadrant I with vertices (0, b), (a, 0), and (0, 0). D. Center one leg of the triangle on the -ais with vertices (0, a), (0, a), and (b, a). E. Use the -ais as one leg of the triangle with vertices (a, 0), (a, b), and (a + c, 0). Online Homework Hints and Help Etra Practice. Describe the position of a general isosceles trapezoid WXYZ determined b the points W (0, 0), X (a, 0), Y (a c, b), and Z (c, b). Then sketch the trapezoid. Write a coordinate proof for the quadrilateral determined b the points (, 4), B (4, 1), C ( 1, 3), and D ( 3, ). 3. Prove that BCD is a parallelogram. 4. Prove that BCD is a rectangle. Module 103 Lesson 4

8 5. Prove that BCD is a rhombus. 6. Prove that BCD is a square. Prove or disprove each statement about the quadrilateral determined b the points W (, 5), X (5, 5), Y (5, 0), and Z (, 0). 7. Prove that the diagonals are congruent. 8. Prove that the diagonals are perpendicular. 9. Prove that the diagonals bisect each other. 10. Prove that WXYZ is a square. lgebra Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that appl. 11. ( 10, 4), B (, 10), C (4, ), D ( 4, 4) 1. J( 9, 7), K( 4, ), L(3, 3), M(, 8) Module 104 Lesson 4

9 nalze Relationships The coordinates of three vertices of parallelogram BCD are given. Find the coordinates of the fourth point so that the given tpe of figure is formed. 13. (4, ), B ( 5, ), D (4, 4), rectangle 14. ( 5, 5), B (0, 0), C (7, 1), rhombus 15. (0, ), B (4, ), C (0, 6), square 16. (, 1), B ( 1, 5), C ( 5, ), square Paul designed a doghouse to fit against the side of his house. His plan consisted of a right triangle on top of a rectangle. Use the drawing for Eercises Find BD, CE, and BE. E 8 in. 6 in. D 10 in. C B 18. Before building the doghouse, Paul sketched his plan on a coordinate plane. He placed at the origin and _ B on the -ais. Find the coordinates of B, C, D, and E, assuming that each unit of the coordinate plane represents one inch. Module 105 Lesson 4

10 19. Critical Thinking On the National Mall in Washington, D.C., a reflecting pool lies between the Lincoln Memorial and the World War II Memorial. The pool has two 300-foot-long sides and two 150-foot-long sides. Tell what additional information ou need to know in order to determine whether the reflecting pool is a rectangle. (Hint: Remember that ou have to show it is a parallelogram first.) lgebra Write a coordinate proof. 0. The Bushmen in South frica use the Global Positioning Sstem to transmit data about endangered animals to conservationists. The Bushmen have sighted animals at the following coordinates: ( 5, 31.5), ( 3., 31.4), and ( 4, 31.1). Prove that the distance between two of these locations is approimatel twice the distance between two other locations. 1. Two cruise ships leave a port located at P (10, 50). One ship sails to an island located at ( 40, 10), and the other sails to an island located at B (60, 110). Suppose that each unit represents one nautical mile. Find the distance between the ships rounded to the nearest nautical mile. Then show that the triangle formed b using the locations of the ships and the port as its vertices is an isosceles triangle. Image Credits: (t) Image Source/lam; (c) Stefanie van der Vin/Fotolia; (b) icholakov/istockphoto.com Module 106 Lesson 4

11 . parallelogram has vertices at (0, 0), (5, 6), and (10, 0). Which could be the fourth verte of the parallelogram? Choose all that appl.. (5, -6) B. (15, 6) C. (0, -6) D. (10, 6) E. (-5, 6) H.O.T. Focus on Higher Order Thinking 3. Draw Conclusions The diagonals of a parallelogram intersect at (-, 1.5). Two vertices are located at (-7, ) and (, 6.5). Find the coordinates of the other two vertices. 4. nalze Relationships Consider points L (3, -4), M (1, -), and N (5, ). a. Find coordinates for point P so that the quadrilateral determined b points L, M, N, and P is a parallelogram. Is there more than one possibilit? Eplain. b. re an of the parallelograms a rectangle? Wh? 5. Critical Thinking Rhombus OPQR has vertices O (0, 0), P (a, b), Q (a + b, a + b), and R (b, a). Prove the diagonals of the rhombus are perpendicular. Module 107 Lesson 4

12 6. Multi-Step Use coordinates to verif the Trapezoid Midsegment Theorem which states The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases. a. M is the midpoint of QP. _ What are its coordinates? b. N is the midpoint of _ RS. What are its coordinates? P (0, 0) M Q (a, b) R (c, b) N S (c + d, 0) c. Find the slopes of QR, _ PS, MN. What can ou conclude? _ d. Find QR, PS, _ MN. _ Show that MN = 1_ (PS + QR). Lesson Performance Task ccording to the new maor, the shape of Cit Park is downright ugl. While the parks in all of the other towns in the vicinit have nice, regular polgonal shapes, Cit Park is the shape of an irregular quadrilateral. On a coordinate map of the park, the four corners are located at (-3, 4), (5, ), (1, -), and (-5, -4). The maor s chief assistant knows a little mathematics and proposes that a special inner park be created b joining the midpoints of the sides of Cit Park. The assistant claims that the boundaries of the inner park will create a nice, regular polgonal shape, just like the parks in all the other towns. The maor thinks the idea is ridiculous, saing, You can t create order out of chaos. 1. Who was right? Eplain our reasoning in detail.. Irregular quadrilateral BCD is shown here. Points J, K, L, and M are midpoints. a. What must ou show to prove that quadrilateral JKLM is a parallelogram? b. How can ou show this? c. If the adjacent sides of JKLM are perpendicular, what tpe of figure does that make JKLM? J M B D K L C Module 108 Lesson 4