5-8 Applying Special Right Triangles
Objective
Justify and apply properties
of 45°-45°-90° triangles.
Justify and apply properties
of 30°- 60°- 90° triangles.
Lesson Presentation
Lesson Review
Holt Geometry
Objectives
Justify and apply properties of
45°-45°-90° triangles.
Justify and apply properties of
30°- 60°- 90° 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.
Objectives
Justify and apply properties
of 45°-45°-90° triangles.
Justify and apply properties of
30°- 60°- 90° triangles.
Example 1: 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 2: 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.
Check for understanding
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
Simplify.
Check for understanding
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.
Example 4: Craft Application
Jana is cutting a square of material for a
tablecloth. The table’s diagonal is 36 inches. She
wants the diagonal of the tablecloth to be an
extra 10 inches so it will hang over the edges of
the table. What size square should Jana cut to
make the tablecloth? Round to the nearest inch.
Jana needs a 45°-45°-90° triangle with a hypotenuse
of 36 + 10 = 46 inches.
Objectives
Justify and apply properties of
45°-45°-90° triangles.
Justify and apply properties
of 30°- 60°- 90° 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.
Example 5: 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.
Example 6: 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.
Check for understanding
Find the values of x and y.
Give your answers in
simplest radical form.
y = 2(5)
y = 10
Simplify.
Example 7: Using the 30º-60º-90º Triangle Theorem
An ornamental pin is in the shape of
an equilateral triangle. The length of
each side is 6 centimeters. Josh will
attach the fastener to the back along
AB. Will the fastener fit if it is 4
centimeters long?
Step 1 The equilateral triangle is divided into two
30°-60°-90° triangles.
The height of the triangle is the length of the
longer leg.
Example 7 Continued
Step 2 Find the length x of the shorter leg.
6 = 2x
3=x
Hypotenuse = 2(shorter leg)
Divide both sides by 2.
Step 3 Find the length h of the longer leg.
The pin is approximately 5.2 centimeters high.
So the fastener will fit.
Check for understanding
A manufacturer wants to make a
larger clock with a height of 30
centimeters. What is the length of
each side of the frame? Round to the
nearest tenth.
Step 1 The equilateral triangle is divided into two
30º-60º-90º triangles.
The height of the triangle is the length of the longer
leg.
Step 2 Find the length x of the shorter leg.
Rationalize the denominator.
Step 3 Find the length y of the longer leg.
y = 2x
Hypotenuse = 2(shorter leg)
Simplify.
Each side is approximately 34.6 cm.
Lesson Review: Part I
Find the values of the variables. Give your
answers in simplest radical form.
1.
2.
x = 10; y = 20
3.
4.
Lesson Review: 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.