1. Triangles and Coordinate Proof
Lesson 4-7
Triangles and Coordinate Proof

2. Standardized Test Practice:
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.
Find the measures of the base angles of an isosceles triangle if the measure of the vertex angle is 58.
Standardized Test Practice: A
32 B
58 C
61 D
122

3. Standardized Test Practice:
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 –
Find the measures of the base angles of an isosceles triangle if the measure of the vertex angle is 58.
Standardized Test Practice: A
32 B
58 C
61 D
122

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
…. Using the distance formula y 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
= √
= √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 triangle XYZ 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.
X (0, 0)
Z (d, 0)

8. Since triangle XYZ is a right triangle the x-coordinate of Y is 0
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)
X (0, 0)
Z (d, 0)

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

10. 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 y-coordinate for S is the distance from R to S. Since QRS is an isosceles right triangle,
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. 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.
Proof: The coordinates of the midpoint D are
The slope of is
or 1. The slope of or –1,
therefore

13. 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. C

14. Proof: Vertex A is at the origin and B is at (0, 10)
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: Homework: Chapter Review handout
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