Download 5.8A Special Right Triangles

5.8A Special Right Triangles
Objectives:
G.SRT.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the
triangle, leading to definitions of trigonometric ratios for acute angles.
For the Board: You will be able to justify and apply the properties of special right triangles.
Anticipatory Set:
Right triangles whose angle measures are 45-45-90 or 30-60-90 are called special right triangles.
A 45°-45°-90° triangle is a special right triangle because the short leg and
the long leg have the same measure.
The triangle is an isosceles right triangle.
Since the legs are equal, instead of labeling the sides a, b, and c, we can
label them a, a, and c.
45
hypotenuse
(c)
leg
(a)
45
leg
(a)
Example: If the equal sides each measure 4 in., what is the
measure of the hypotenuse? Write the answer using
45
simplified radicals.
4
42 + 42 = c2
16 + 16 = c2
32 = c2
c = 32  4 2
c
45
4
Practice: If the equal sides each measure 7 ft., what is the
measure of the hypotenuse? Write the answer using
simplified radicals.
72 + 72 = c2
49 + 49 = c2
98 = c2
c = 98  7 2
45
c
7
45
7
45-45-90 Triangle Theorem
In a 45-45-90 triangle, a 2  c.
45
Example: If the hypotenuse measures 12 cm, what are the
measures of the equal sides? Write the answer
using simplified radicals.
a2 + a2 = 122
2a2 = 144
a2 = 72
a = 72  6 2
Practice: If the hypotenuse measures 25 mm, what are the
measures of the equal sides? Write the answer
using simplified radicals.
a2 + a2 = 282
2a2 = 784
a2 = 392
a = 392  14 2
45
12
a
28
a
45
a
45
a
45-45-90 Triangle Theorem
In a 45-45-90 triangle, a 
1
c 2
2
When working with 45-45-90 triangles the answers should be written as simplified radicals unless
otherwise stated.
Example: Find the unknown sides.
Note: x and y will always have the same value.
1.
2.
3.
4.
x = 5, find y and z.
y = 5 and z = 5 2
y = 14 find x and z.
x = 14 and z = 14 2
z = 7 2 , find x and y.
x = 7 and y = 7
z = 12 6 , find x and y.
x 2 = 12 6 so x = 12 3 and y = 12 3
18
5. z = 18, find x and y.
18 = x 2
x
2
18
2

2
2

18 2
4
White Board Practice:
Practice: Find the unknown sides.
1. x = 7, find y and z.
2. y = 22, find x and z.
3. z = 5 2 , find x and y
4. z = 16 10 , find x and y.
5. z = 24, find x and y.
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 372 – 374 prob. 1 – 4, 9 – 12.
For a Grade:
Text: pgs. 372 – 374 prob. 2, 4, 10, 12.

18 2
9 2 a
2
45
z
x
45
y
x = 9 2 and y = 2
45
y = 7, z = 7 2
x = 22, z = 22 2
x 2 = 5 2 so x = 5, y = 5
x 2 = 16 10 so x = 16 5
x 2 = 25 so x =
z
x
45
y
24
24 2 24 2
,x=
=
 12 2
2
2
2 2