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Holt McDougal Geometry
5-8 Applying Special Right Triangles 5-8 Applying Special Right Triangles
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
Holt McDougal Geometry
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.
Holt McDougal 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
Holt McDougal Geometry
5-8 Applying Special Right Triangles
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.
Holt McDougal Geometry
5-8 Applying Special Right Triangles
Holt McDougal Geometry
5-8 Applying Special Right Triangles
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.
Holt McDougal Geometry
5-8 Applying Special Right Triangles
Example 1B: Finding Side Lengths in a 45º- 45º- 90º
Triangle
Find the value of x. Give your answer in simplest radical form.
The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 5.
Rationalize the denominator.
Holt McDougal Geometry
5-8 Applying Special Right Triangles
Check It Out! Example 1a
Find the value of x. Give your answer in simplest radical form.
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
Holt McDougal Geometry
5-8 Applying Special Right Triangles
Check It Out! Example 1b
Find the value of x. Give your answer in simplest radical form.
The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 16.
Rationalize the denominator.
Holt McDougal Geometry
5-8 Applying Special Right Triangles
A 30°-60°-90° triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths.
Holt McDougal Geometry
5-8 Applying Special Right Triangles
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.
Holt McDougal Geometry
5-8 Applying Special Right Triangles
Example 3B: Finding Side Lengths in a 30º-60º-90º
Triangle
Find the values of x and y. Give your answers in simplest radical form.
Rationalize the denominator.
Hypotenuse = 2(shorter leg).
Simplify.
y = 2x
Holt McDougal Geometry
5-8 Applying Special Right Triangles
Check It Out! Example 3a
Find the values of x and y. Give your answers in simplest radical form.
Hypotenuse = 2(shorter leg)
Divide both sides by 2.
y = 27 Substitute for x.
Holt McDougal Geometry
5-8 Applying Special Right Triangles
Check It Out! Example 3b
Find the values of x and y. Give your answers in simplest radical form.
Simplify.
y = 2(5)
y = 10
Holt McDougal Geometry
5-8 Applying Special Right Triangles
Check It Out! Example 3c
Find the values of x and y. Give your answers in simplest radical form.
Hypotenuse = 2(shorter leg)
Divide both sides by 2.
Substitute 12 for x.
24 = 2x
12 = x
Holt McDougal Geometry
5-8 Applying Special Right Triangles
Check It Out! Example 3d
Find the values of x and y. Give your answers in simplest radical form.
Rationalize the denominator.
Hypotenuse = 2(shorter leg) x = 2y
Simplify.