applying special right trianglesapplying special …30 - 60 - 90 triangles. objectives holt mcdougal...

Holt McDougal Geometry

5-8 Applying Special Right Triangles 5-8 Applying Special Right Triangles

Holt Geometry

Warm Up

Lesson Presentation

Lesson Quiz



5-8 Applying Special Right Triangles

Warm Up For Exercises 1 and 2, find the value of x. Give your answer in simplest radical form.

1. 2.

Simplify each expression.

3. 4.



Justify and apply properties of 45°-45°-90° triangles.

Justify and apply properties of 30°- 60°- 90° triangles.

Objectives



A diagonal of a square divides it into two congruent isosceles right triangles. Since the base angles of an isosceles triangle are congruent, the measure of each acute angle is 45°. So another name for an isosceles right triangle is a 45°-45°-90° triangle.

A 45°-45°-90° triangle is one type of special right triangle. You can use the Pythagorean Theorem to find a relationship among the side lengths of a 45°-45°-90° triangle.



Example 1A: Finding Side Lengths in a 45°- 45º- 90º

Triangle

Find the value of x. Give your answer in simplest radical form.

By the Triangle Sum Theorem, the measure of the third angle in the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of 8.



Example 1B: Finding Side Lengths in a 45º- 45º- 90º

Triangle


The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 5.

Rationalize the denominator.



Check It Out! Example 1a


x = 20 Simplify.

By the Triangle Sum Theorem, the measure of the third angle in the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of



Check It Out! Example 1b


The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 16.




A 30°-60°-90° triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths.



Example 3A: Finding Side Lengths in a 30º-60º-90º

Triangle

Find the values of x and y. Give your answers in simplest radical form.

Hypotenuse = 2(shorter leg) 22 = 2x

Divide both sides by 2. 11 = x

Substitute 11 for x.



Example 3B: Finding Side Lengths in a 30º-60º-90º

Triangle



Hypotenuse = 2(shorter leg).

Simplify.

y = 2x



Check It Out! Example 3a


Hypotenuse = 2(shorter leg)

Divide both sides by 2.

y = 27 Substitute for x.



Check It Out! Example 3b


Simplify.

y = 2(5)

y = 10



Check It Out! Example 3c


Hypotenuse = 2(shorter leg)

Divide both sides by 2.

Substitute 12 for x.

24 = 2x

12 = x



Check It Out! Example 3d



Hypotenuse = 2(shorter leg) x = 2y

Simplify.

applying special right trianglesapplying special …30 - 60 - 90 triangles. objectives holt mcdougal...

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