Download 8-3 Special Right Triangles

8-3 Special Right
Triangles
GEOMETRY LESSON 8-2
Special Right Triangles
(For help, go to Lesson 1-6.)
Use a protractor to find the measures of the angles of each triangle.
1.
2.
3.
Check Skills You’ll Need
8-3
Special Right Triangles
GEOMETRY LESSON 8-2
Solutions
1. 45, 45, 90
2. 30, 60, 90
3. 45, 45, 90
8-3
GEOMETRY LESSON 8-2
Special Right Triangles
Find the length of the hypotenuse of a 45°-45°-90°
triangle with legs of length 5 6.
Use the 45°-45°-90° Triangle Theorem to find the hypotenuse.
h=
2•5 6
h = 5 12
hypotenuse = 2 • leg
Simplify.
h = 5 4(3)
h = 5(2)
3
h = 10 3
The length of the hypotenuse is 10
3.
Quick Check
8-3
GEOMETRY LESSON 8-2
Special Right Triangles
Find the length of a leg of a 45°-45°-90° triangle with a
hypotenuse of length 22.
Use the 45°-45°-90° Triangle Theorem to find the leg.
22 = 2 • leg
hypotenuse =
x = 22
Divide each side by
2
x=
22
•
2
x=
22 2
2
x = 11 2
2
2
2 • leg
2.
Simplify by rationalizing the
denominator.
Simplify.
The length of the leg is 11 2.
Quick Check
8-3
GEOMETRY LESSON 8-2
Special Right Triangles
The distance from one corner to the opposite corner of a
square playground is 96 ft. To the nearest foot, how long is each side of
the playground?
The distance from one corner to the opposite corner, 96 ft, is the length
of the hypotenuse of a 45°-45°-90° triangle.
96 =
2 • leg
hypotenuse =
leg =
96
2
Divide each side by
leg = 67.882251
2 • leg
2.
Use a calculator.
Each side of the playground is about 68 ft.
Quick Check
8-3
GEOMETRY LESSON 8-2
Special Right Triangles
Quick Check
The longer leg of a 30°-60°-90° triangle has
length 18. Find the lengths of the shorter leg and the hypotenuse.
You can use the 30°-60°-90° Triangle Theorem to find the lengths.
18 = 3 • shorter leg
d=
18
3
d=
18
• 3
3
3
d = 18
longer leg =
3 • shorter leg
Divide each side by
3.
Simplify by rationalizing
the denominator.
3
3
d=6 3
Simplify.
f=2•6 3
hypotenuse = 2 • shorter leg
f = 12 3
Simplify.
The length of the shorter leg is 6 3, and the length of the hypotenuse is 12 3.
8-3
Special Right Triangles
GEOMETRY LESSON 8-2
A garden shaped like a rhombus has a perimeter of 100 ft and
a 60° angle. Find the area of the garden to the nearest square foot.
Because a rhombus has four sides of equal length, each side is 25 ft.
Because a rhombus is also a parallelogram, you can use the area
formula A = bh.
Draw the rhombus with altitude h,
and then solve for h.
8-3
GEOMETRY LESSON 8-2
Special Right Triangles
(continued)
The height h is the longer leg of the right triangle. To find the height h,
you can use the properties of 30°-60°-90° triangles. Then apply
the area formula.
25 = 2 • shorter leg
25
shorter leg = 2 = 12.5
hypotenuse = 2 • shorter leg
Divide each side by 2.
h = 12.5 3
longer leg = 3 • shorter leg
A = bh
Use the formula for the area of a parallelogram.
A = (25)(12.5 3) Substitute 25 for b and 12.5 3 for h.
A = 541.26588
Use a calculator.
To the nearest square foot, the area is 541 ft2.
8-3
Quick Check
GEOMETRY LESSON 8-2
Use
Special Right Triangles
ABC for Exercises 1–3.
1. If m
A = 45, find AC and AB.
AC = 18; AB = 18 2
2. If m A = 30, find AC and AB.
AC = 18 3; AB = 36
3. If m A = 60, find AC and AB.
AC = 6 3; AB = 12 3
4. Find the side length of a 45°-45°-90° triangle with a 4-cm
hypotenuse.
2 2 , or about 2.8 cm
5. Two 12-mm sides of a triangle form a 120° angle. Find the length
of the third side.
12 3, or about 20.8 mm
8-3