SPECIAL RIGHT TRIANGLES

1

Chapter 7 Lesson 2
SPECIAL RIGHT TRIANGLES
Objectives 1. Use Properties of 45 45 90
triangles 2. Use Properties of 30 60 90
triangles
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45- 45 - 90 TRIANGLE

• x2 x2 h2
• 2x2 h2
• The length of the hypotenuse of any 45 -45 -90
  triangle is times the length of its leg.
• The ratio of the sides is

h
x
x
3
EXAMPLE 1

• Find x.
• leg 3
• hypotenuse leg
• x
• x

x
3
3
4
EXAMPLE 2

• Find x.
• x

5
5
x
5
EXAMPLE 3

• Find x
• hypotenuse
• leg hyp.
• x 7

x
6
EXAMPLE 4

• Find x
• x

18
x
x
7
EXAMPLE 5

• Find the length of the diagonal of a square whose
  side is 10 m.
• x m

x
10 m
8
30- 60- 90 TRIANGLE

• When an altitude is drawn from any vertex of an
  equilateral triangle, two congruent 30 -60 -90
  triangles are formed.
• a2 x2 (2x)2
• a2 x2 4x2
• a2 3x2
• a x

60
30
30
2x
2x
a
60
60
x
x
2x
9

• In a 30-60-90 triangle the measures of the
  sides are x, x and 2x.
• The ratio of the sides is

30
2x
60
x
10
30-60-90 TRIANGLE
30
2x

• In a nutshell
• hypotenuse 2 times short leg 2x
• short leg times hypotenuse
• long leg times short leg x

hypotenuse
long leg
60
x
short leg
11
EXAMPLE 6

• Find x and y.
• x is the short leg
• x hypotenuse
• x
• x 8
• y is the long leg
• y short leg
• y 8

30
16 in.
y
60
x
12
EXAMPLE 7

• Find x and y.
• x 6
• y

12
y
60
x
13
EXAMPLE 8

• Find x and y.
• X 10
• Y

60
x
5
y
14
EXAMPLE 9

• Find x and y.
• x
• y 18

x
30
y
15
EXAMPLE 10

• Find x and y.
• x 4
• y 8

30
x
y
16
EXAMPLE 11
24

• Find x and y.

30
x
y
17
EXAMPLE 11
24

• Find x and y.
• x

30
x
y
18
EXAMPLE 11
24

• Find x and y.
• x
• y

30
x
y
19
REMEMBER

• In a 45-45-90 triangle
• hyp. leg
• leg hyp.

• In a 30-60-90 triangle
• hyp. short leg
• short leg hyp.
• long leg short leg
• short leg long leg