Lesson 4-7

Triangles and Coordinate Proof

5-Minute Check on Lesson 4-65-Minute Check on Lesson 4-65-Minute Check on Lesson 4-65-Minute Check on Lesson 4-6 Transparency 4-7

Refer to the figure.

1. Name two congruent segments if 1 2.

2. Name two congruent angles if RS RT.

3. Find mR if mRUV = 65.

4. Find mC if ABC is isosceles with AB AC and mA = 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:

A CB D32 58 61 122

5-Minute Check on Lesson 4-65-Minute Check on Lesson 4-65-Minute Check on Lesson 4-65-Minute Check on Lesson 4-6 Transparency 4-7

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 mR if mRUV = 65. 50

4. Find mC if ABC is isosceles with AB AC and mA = 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:

A CB D32 58 61 122

Objectives

• Position and label triangles for use in coordinate proofs

• Write coordinate proofs

Vocabulary

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

Classifying Triangles

…. Using the distance formula

Find the measures of the sides of ▲DEC.Classify the triangle by its sides.

D (3, 9) E (3, -5) C (2, 2)

E

D

C

y

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 = 10DC = √ (3 – 2)2 + (9 – 2)2

= √(1)2 + (7)2

= √1 + 49 = √50

DC = EC, so ▲DEC is isosceles

Use the origin as vertex X of the triangle.

Place the base of the triangle along the positive x-axis.

Position and label right triangle XYZ with leg d units long on the coordinate plane.

X (0, 0) Z (d, 0)

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.

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:

X (0, 0) Z (d, 0)

Y (0, b)

Answer:

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

Name the missing coordinates of isosceles right triangle QRS.

Answer: Q(0, 0); S(c, c)

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: C(0, 0); A(0, d)

Name the missing coordinates of isosceles right ABC.

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 .

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

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.

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