Two Special Right Triangles
45°- 45°- 90°
30°- 60°- 90°
HW: Special Right Triangles WS1
(side 1 only: 45-45-90)
45°- 45°- 90°
1
1
1
1
The 45-45-90
triangle is
based on the
square with
sides of 1 unit.
45°- 45°- 90°
1
45°
1
45°
1
45°
45°
1
If we draw the
diagonals we
form two
45-45-90
triangles.
45°- 45°- 90°
1
45°
1
45°
1
45°
45°
1
Using the
Pythagorean
Theorem we can
find the length of
the diagonal.
45°- 45°- 90°
1
45°
1
45°
2
1
45°
45°
1
45°- 45°- 90°
45°
1
45°
1
In a 45° – 45° – 90° triangle
the hypotenuse is the
square root of two times as
long as each leg
Rule:
45°- 45°- 90° Practice
45°
4
45°
SAME
4
45°- 45°- 90° Practice
45°
9
45°
SAME
9
45°- 45°- 90° Practice
45°
45°
SAME
45°- 45°- 90° Practice
45°- 45°- 90° Practice
45°
45°
45°- 45°- 90° Practice
=3
45°- 45°- 90° Practice
45°
3
45°
SAME
3
45°- 45°- 90° Practice
45°
45°
45°- 45°- 90° Practice
= 11
45°- 45°- 90° Practice
45°
11
45°
SAME
11
45°- 45°- 90° Practice
45°
8
45°
45°- 45°- 90° Practice
8
*
=
2
Rationalize the denominator
45°- 45°- 90° Practice
45°
8
45°
SAME
45°- 45°- 90° Practice
45°
4
45°
45°- 45°- 90° Practice
4
*
=
2
Rationalize the denominator
45°- 45°- 90° Practice
45°
4
45°
SAME
45°- 45°- 90° Practice
45°
7
45°
45°- 45°- 90° Practice
7
*
Rationalize the denominator
45°- 45°- 90° Practice
45°
7
45°
SAME
Find the value of each variable.
Write answers in simplest radical form.
Find the value of each variable.
Write the answers in simplest radical form.
• Know the basic
triangles
• Set known information
equal to the
corresponding part of
the basic triangle
• Solve for the other
sides
10  2 x x  5 2 y  5 2
Find the value of each variable. Write
answers in simplest radical form.
Two Special Right Triangles
45°- 45°- 90°
30°- 60°- 90°
HW: Special Right Triangles WS1
(side 2 only: 30-60-90)
30°- 60°- 90°
2
2
60°
60°
2
The 30-60-90
triangle is based
on an
equilateral
triangle with
sides of 2 units.
30°- 60°- 90°
2 30° 30°
60°
1
2
60°
2
1
The altitude cuts
the triangle into
two congruent
triangles.
Long Leg
30°- 60°- 90°
This creates
30°
the 30-60-90
triangle with a
hypotenuse
a
60°
short
leg
and
Short Leg
a long leg.
30°- 60°- 90° Practice
We saw that the
hypotenuse is twice
the short leg.
30°
2
60°
1
We can use the
Pythagorean
Theorem to find
the long leg.
30°- 60°- 90° Practice
30°
2
60°
1
30°- 60°- 90°
30°
2
60°
1
30° – 60° – 90°
Triangle
In a 30° – 60° – 90°
triangle, the
hypotenuse is twice
as long as the
shorter leg, and the
longer leg is the
square root of three
times as long as the
shorter leg
30°-60°-90°
30°- 60°- 90° Practice
30°
8
60°
4
The key is to find
the length of the
short side.
Hypotenuse =
short leg * 2
30°- 60°- 90° Practice
30°
10
60°
5
hyp =
short leg * 2
30°- 60°- 90° Practice
30°
14
60°
7
*2
30°- 60°- 90° Practice
30°
3
60°
*2
30°- 60°- 90° Practice
30°
60°
*2
30°- 60°- 90° Practice
30°- 60°- 90° Practice
30°
22
60°
11
Short Leg =
hyp 2
30°- 60°- 90° Practice
30°
4
60°
2
30°- 60°- 90° Practice
30°
18
60°
9
30°- 60°- 90° Practice
30°
46
60°
23
30°- 60°- 90° Practice
30°
28
60°
14
30°- 60°- 90° Practice
30°
9
60°
30°- 60°- 90° Practice
30°
12
60°
hyp =
Short Leg * 2
30°- 60°- 90° Practice
30°
27
60°
hyp =
Short Leg * 2
30°- 60°- 90° Practice
30°
20
60°
hyp =
Short Leg * 2
30°- 60°- 90° Practice
30°
33
60°
hyp =
Short Leg * 2
PRACTICE
Find all the missing sides for
each triangle.
Solving Strategy
•
•
•
Know the basic triangles
Set known information equal to the
corresponding part of the basic triangle
Solve for the other sides
Find the value of each variable. Write
answers in simplest radical form.
Find the value of each variable. Write
answers in simplest radical form.
12
60
24
30
12 3
60
30
5.5
5.5 
5.5
2
10
10
10
2
1
60
2
30
3
10
10
5
10
60
30
8
8 2
30
60
3
60
6
30
3 3
15
15
15 2
15
15
60
8 3
4 3
30
12
8
2
2
2
18
18
9 3
18
Find the distance across the canyon.
30-60-90
b
Find the length of the canyon wall (from
the edge to the river).
b
c=b*2
c
Is it more or less than a mile across the
canyon?
5280 ft = 1 mile