Understanding Triangles and Coordinate Proofs in Geometry

1. Lesson 4-7 Triangles and Coordinate Proof

2. Transparency 4-7 5-Minute Check on Lesson 4-6 Refer to the figure. 1. Name two congruent segments if 1 2. 2. Name two congruent angles if RSRT. 3. Find mR if mRUV = 65. 4. Find mC if ABC is isosceles with ABAC and mA = 70. 5. Find x if LMN is equilateral with LM = 2x – 4, MN = x + 6,and LN = 3x – 14. 6. Find the measures of the base angles of an isosceles triangle if the measure of the vertex angle is 58. Standardized Test Practice: 122 D C 61 A 32 B 58

3. Transparency 4-7 5-Minute Check on Lesson 4-6 Refer to the figure. 1. Name two congruent segments if 1 2. UWVW 2. Name two congruent angles if RSRT. S T 3. Find mR if mRUV = 65. 50 4. Find mC if ABC is isosceles with ABAC and mA = 70. 55 5. Find x if LMN is equilateral with LM = 2x – 4, MN = x + 6,and LN = 3x – 14. 10 6. Find the measures of the base angles of an isosceles triangle if the measure of the vertex angle is 58. Standardized Test Practice: 122 D C 61 A 32 B 58

4. Objectives • Position and label triangles for use in coordinate proofs • Write coordinate proofs

5. Vocabulary • Coordinate proof – uses figures in the coordinate plane and algebra to prove geometric concepts.

6. Classifying Triangles y …. Using the distance formula D Find the measures of the sides of ▲DEC. Classify the triangle by its sides. D (3, 9) E (3, -5) C (2, 2) E C x EC = √ (-5 – 2)2 + (3 – 2)2 =√(-7)2 + (1)2 = √49 + 1 = √50 ED = √ (-5 – 3)2 + (3 – 9)2 = √(-8)2 + (-6)2 = √64 + 36 = √100 = 10 DC = √ (3 – 2)2 + (9 – 2)2 =√(1)2 + (7)2 = √1 + 49 = √50 DC = EC, so ▲DEC is isosceles

7. Position and label right triangleXYZ with leg d units long on the coordinate plane. Use the origin as vertex X of the triangle. Place the base of the triangle along the positive x-axis. Position the triangle in the first quadrant. Since Z is on the x-axis, its y-coordinate is 0. Its x-coordinate is d because the base is d units long. Z (d, 0) X (0, 0)

8. Since triangle XYZ is a right triangle the x-coordinate of Y is 0. We cannot determine the y-coordinate so call it b. Answer: Y (0, b) Z (d, 0) X (0, 0)

9. Position and label equilateral triangleABC with side w units long on the coordinate plane. Answer:

10. The y-coordinate for S is the distance from R to S. Since QRS is an isosceles right triangle, Name the missing coordinates of isosceles right triangle QRS. Q is on the origin, so its coordinates are (0, 0). The x-coordinate of S is the same as the x-coordinate for R, (c, ?). The distance from Q to R is c units. The distance from R to S must be the same. So, the coordinates of S are (c, c). Answer: Q(0, 0); S(c, c)

11. Name the missing coordinates of isosceles right ABC. Answer: C(0, 0); A(0, d)

12. Proof: The coordinates of the midpoint D are The slope of is or 1. The slope of or –1, therefore . Write a coordinate proof to prove that the segment drawn from the right angle to the midpoint of the hypotenuse of an isosceles right triangle is perpendicular to the hypotenuse.

13. C FLAGS Write a coordinate proof to prove this flag is shaped like an isosceles triangle. The length is 16 inches and the height is 10 inches.

14. Proof: Vertex A is at the origin and B is at (0, 10). The x-coordinate of C is 16. The y-coordinate is halfway between 0 and 10 or 5. So, the coordinates of C are (16, 5). Determine the lengths of CA and CB. Since each leg is the same length, ABC is isosceles. The flag is shaped like an isosceles triangle.

15. Summary & Homework • Summary: • Coordinate proofs use algebra to prove geometric concepts. • The distance formula, slope formula, and midpoint formula are often used in coordinate proofs. • Homework: Chapter Review handout