5-8
5-8 Applying
ApplyingSpecial
SpecialRight
RightTriangles
Triangles
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
GeometryGeometry
Holt
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.
Holt McDougal Geometry
4.
5-8 Applying Special Right Triangles
Objectives
Justify and apply properties of
45°-45°-90° triangles.
Justify and apply properties of
30°- 60°- 90° triangles.
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.
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
x = 20
Holt McDougal Geometry
Simplify.
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.
22 = 2x
Hypotenuse = 2(shorter leg)
11 = x
Divide both sides by 2.
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.
y = 2x
Hypotenuse = 2(shorter leg).
Simplify.
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
Holt McDougal Geometry
Substitute
for x.
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.
y = 2(5)
y = 10
Simplify.
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.
24 = 2x
Hypotenuse = 2(shorter leg)
12 = x
Divide both sides by 2.
Substitute 12 for 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.
x = 2y
Hypotenuse = 2(shorter leg)
Simplify.
Holt McDougal Geometry