To prove theorems using figures in the coordinate plane

Transcription

1 6-9 s Using Coordinate Geometr Content Standard G.GPE.4 Use coordinates to prove simple geometric theorems algebraicall. bjective To prove theorems using figures in the coordinate plane Better draw a diagram! The coordinates of three vertices of a rectangle are ( a, 0), (a, 0), and (a, b). A diagonal joins one of these points with the fourth verte. What are the coordinates of the midpoint of the diagonal? Justif our answer. ATHEATICAL In the Solve It, the coordinates of the point include variables. In this lesson, ou will use coordinates with variables to write a coordinate proof. Essential Understanding You can prove geometric relationships using variable coordinates for figures in the coordinate plane. What formulas do ou need? You need to find the distance to a midpoint, so use the midpoint and distance formulas. Problem 1 Writing a Coordinate Use coordinate geometr to prove that the midpoint of the hpotenuse of a right triangle is equidistant from the three vertices. Given: EF is a right triangle. is the midpoint of EF. Prove: E F Coordinate : a 0 0 b B the idpoint Formula,, (a, b). B the Distance Formula, a b E(0, b) F (a a) (0 b) E (0 a) (b b) a b a b F(a, 0) Since E F, the midpoint of the hpotenuse is equidistant from the vertices of the right triangle. Got It? 1. Reasoning What is the advantage of using coordinates (0, 0), E(0, b), and F(a, 0) rather than (0, 0), E(0, b), and F(a, 0)? 414 Chapter 6 Polgons and Quadrilaterals

2 In Lesson 6-8, ou wrote a plan for the proof of the Trapezoid idsegment Theorem. ow ou will write the full coordinate proof. Problem Writing a Coordinate Refer to the plan from Lesson 6-8. Find the coordinates of and. Determine whether is parallel to P and RA. Then find and compare the lengths of, P, and RA. Write a coordinate proof of the Trapezoid idsegment Theorem. Given: Prove: is the midsegment of trapezoid RAP. P, RA, 1 (P RA) Coordinate : Use the idpoint Formula to find the coordinates of and. R(b, c) A(d, c) P(a, 0) b 0, c 0 (b, c) a d, 0 c (a d, c) Use the Slope Formula to determine whether is parallel to P and RA. c c slope of (a d) b 0 c c slope of RA d b 0 slope of P 0 0 a 0 0 The three slopes are equal, so P and RA. Use the Distance Formula to find and compare, P, and RA. (a d) b) (c c) a d b P (a 0) (0 0) a RA (d b) (c c) d b 1 (P RA) Check a d b 1 a (d b) Substitute. a d b a d b that 1 (P RA) is true. Simplif. So, (1) the midsegment of a trapezoid is parallel to its bases, and () the length of the midsegment of a trapezoid is half the sum of the lengths of the bases. Got It?. Write a coordinate proof of the Triangle idsegment Theorem (Theorem 5-1). Lesson 6-9 s Using Coordinate Geometr 415

3 Lesson Check Do ou know HW? 1. Use coordinate geometr to prove that the diagonals of a rectangle are congruent. a. Place rectangle PQRS in the coordinate plane with P at (0, 0). b. What are the coordinates of Q, R, and S? c. Write the Given and Prove statements. d. Write a coordinate proof. Do ou UDERSTAD?. Reasoning Describe a good strateg for placing the vertices of a rhombus for a coordinate proof. 3. Error Analsis Your classmate places a trapezoid on the coordinate plane. What is the error? ATHEATICAL Practice and Problem-Solving Eercises ATHEATICAL A Practice Developing Complete the following coordinate proofs. See Problems 1 and. 4. The diagonals of an isosceles trapezoid are congruent. Given: Trapezoid EFGH with EF GH Prove: EG FH a. Find EG. b. Find FH. c. Eplain wh EG FH. F( b, c) G(b, c) E( a, 0) H(a, 0) B Appl 5. The medians drawn to the congruent sides of an isosceles triangle are congruent. Given: PQR with PQ RQ, is the midpoint of PQ, is the midpoint of RQ Prove: P R a. What are the coordinates of and? b. What are P and R? c. Eplain wh P R. Tell whether ou can reach each tpe of conclusion below using coordinate methods. Give a reason for each answer. 6. AB CD 7. AB CD 8. AB CD 9. AB bisects CD. 10. AB bisects CAD. 11. A B 1. A is a right angle. 13. AB BC AC 14. ABC is isosceles. 15. Quadrilateral ABCD is a rhombus. 16. AB and CD bisect each other. 17. A is the supplement of B. 18. AB, CD, and EF are concurrent. Q(0, b) P( a, 0) R(a, 0) 416 Chapter 6 Polgons and Quadrilaterals

4 C Challenge 19. Flag Design The flag design at the right is made b connecting the midpoints of the sides of a rectangle. Use coordinate geometr to prove that the quadrilateral formed is a rhombus. 0. pen-ended Give an eample of a statement that ou think is easier to prove with a coordinate geometr proof than with a proof method that does not require coordinate geometr. Eplain our choice. Use coordinate geometr to prove each statement. 1. Think About a Plan If a parallelogram is a rhombus, its diagonals are perpendicular (Theorem 6-13). How will ou place the rhombus in a coordinate plane? What formulas will ou need to use?. The altitude to the base of an isosceles triangle bisects the base. 3. If the midpoints of a trapezoid are joined to form a quadrilateral, then the quadrilateral is a parallelogram. 4. ne diagonal of a kite divides the kite into two congruent triangles. 5. You learned in Theorem 5-8 that the centroid of a triangle is two thirds the distance from each verte to the midpoint of the opposite side. Complete the steps to prove this theorem. a. Find the coordinates of points L,, and, the midpoints of the sides of ABC. b. Find equations of A, B, and CL. c. Find the coordinates of point P, the intersection of A and B. d. Show that point P is on CL. e. Use the Distance Formula to show that point P is two thirds the distance from each verte to the midpoint of the opposite side. 6. Complete the steps to prove Theorem 5-9. You are given ABC with altitudes p, q, and r. Show that p, q, and r intersect at a point (called the orthocenter of the triangle). a. The slope of BC is c b. What is the slope of line p? b. Show that the equation of line p is b c ( a). c. What is the equation of line q? d. Show that lines p and q intersect at 0, ab c. c e. The slope of AC is a. What is the slope of line r? f. Show that the equation of line r is a c ( b). g. Show that lines r and q intersect at 0, ab c. h. What are the coordinates of the orthocenter of ABC? 7. ultiple Representations Use the diagram at the right. a. Eplain using area wh 1 ad 1 bc and therefore ad bc. b. Find two ratios for the slope of. Use these two ratios to show that ad bc. A r A(a, 0) L d B(6q, 6r) c b P q C(0, c) a C(6p, 0) p p B(b, 0) Lesson 6-9 s Using Coordinate Geometr 417

5 8. Prove: If two lines are perpendicular, the product of their slopes is 1. a. Two nonvertical lines, 1 and, intersect as shown at the right. Find the coordinates of C. b. Choose coordinates for D and B. (Hint: Find the relationship between 1,, and 3. Then use congruent triangles.) c. Complete the proof that the product of slopes is 1. B(, ) D(, ) 3 A(a, b) 1 C(, ) 1 Standardized Test Prep SAT/ACT 9. The endpoint of a segment is (7, 3). The midpoint is (3, 4). What is the length of the segment? 30. In the diagram of PR at the right, P 16 and R 1. What is? 16 P 31. FGHI has sides with lengths FG 5, GH 7, HI 3, and FI. What is the length of the longer sides of FGHI? 3. In ABC, m A 55. If m C is twice m A, what is m B? 8 8 R 16 ied Review 33. A rectangle ABCD is centered at the origin with A( a, b). Without using an new variables, what are the coordinates of point C? Eplain how ou can use SSS, SAS, ASA, or AAS with corresponding parts of congruent triangles to prove each statement true. See Lesson 6-8. See Lessons 4-, 4-3, and AB CB K A C E L B H 1 F D G K Get Read! To prepare for Lesson 7-1, do Eercises Algebra Solve. Round to the nearest tenth if necessar. See p r k Chapter 6 Polgons and Quadrilaterals