Holt Geometry 5-8 Applying Special Right Triangles Justify and apply properties of 45°-45°-90°...

Holt Geometry

5-8 Applying Special Right Triangles

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

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

Objectives

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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.

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Holt Geometry


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.

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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.

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Example 1c


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

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Example 1d


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


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

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Example 2A: 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.

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Example 2B: Finding Side Lengths in a 30º-60º-90º Triangle



Hypotenuse = 2(shorter leg).

Simplify.

y = 2x

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Example 2c


Hypotenuse = 2(shorter leg)

Divide both sides by 2.

y = 27 Substitute for x.

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Check It Out! Example 2d


Simplify.

y = 2(5)

y = 10

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Lesson Quiz: Part IFind the values of the variables. Give your answers in simplest radical form.

1. 2.

3. 4.

x = 10; y = 20

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Lesson Quiz: Part II

Find the perimeter and area of each figure. Give your answers in simplest radical form.

5. a square with diagonal length 20 cm

6. an equilateral triangle with height 24 in.

Holt Geometry 5-8 Applying Special Right Triangles Justify and apply properties of 45°-45°-90°...

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