1. Special Cases, for Right Triangles
60°
30°
45°
45° 45° 90° Triangles
45°

2. 30° 60° 90° triangles have just been born
1. An equilateral triangle is also equiangular, all angles are the same.
60° 2
2. Let’s draw an Altitude from one of the vertices Which is also a Median and Angle bisector. 1
30° 2
60°
30°
3. The bisected side is divided into two equal segments and the bisected angle has now two 30° equal angles. 2
60°
Congratulations! Two
30° 60° 90° triangles have just been born
Oooh …and you watched!!!!

3. And the other leg is times as big, 1 * =
Let’s separate the top triangle and label the unknown side as z. 1
60°
30° 2
60°
30° 2
60°
apply the Pythagorean Theorem to find the unknown side. 1 1 2
2 = z + 1 2
4 = z 2
z = 3 z
3 = z 2
3 = z 2
When, the smallest side is equal to 1, the hypotenuse is 2 times as big, 1 * 2 = 2
And the other leg is times as big,
1 * =
Can we generalize this result for all °-60°-90° right triangles?
z = 3

4. 3 3 3 60° 30° 1 2 60° (2) 4 2 (2) 2 1 Yes it works! 30° (2) 1 2 (.5)
Is this true for a triangle that is twice as big?
Is this true for a triangle that is half the original size?
Yes , it still works.
If we know 1 side length of a triangle,
we can use this pattern to find the other 2 sides

5. the hypotenuse is twice as long as the shorter leg, and 3 s
60° 2
30°
In a 30°-60°-90° triangle,
the hypotenuse is twice as long as the shorter leg, and
the longer leg is times as long as the shorter leg. 3

6. Find the values of the variables
Find the values of the variables. Round your answers to the nearest hundredth. y
2x = 14
30°
2x = 14 x 14
60°
x =2
Is this 30°-60°-90°?
90°-30°=60°
y = x 3
Then we know that:
y = 2 3 2 3 s
60°
30°

7. . 90 = x 3 2x = y 90 = x 3 ( ) = y 2 3 3 y 90 = y 3 90 3 = x 3 y= 3 OR
Find the values of the variables. Round your answers to the nearest unit.
90 = x 3
2x = y
90 = x 3
( )
= y 2 3 30
30° 3 y 90
= y 3 60 . 90 3
= x 3 y= 3 60 OR
60° 3 90
= x x 3
( ) 2
y =104
Is this a 30°-60°-90°?
90°-60°=30° 3 90
= x 2 3 s
60°
30° 3 30
= x 3 30 x=
x = 52 OR

8. Find the values of the variables. Find the exact answer.3
2x = y
60°
30 = x 3
( )
= y 2 3 10 3
= y 3 20 y . 30 3
= x 3 30 y= 3 20
30° 3 30
= x 3
( ) 2
Is this a 30°-60°-90°?
90°-60°=30° 3 30
= x 2 3 s
60°
30° 3 10
= x 3 10 x=

9. What kind of right triangles are formed?1
Let’s draw a diagonal for the square. The diagonal bisects the right angles of the square.
What kind of right triangles are formed?

10. 1
45° 1
45°
45° y
y = 2
The triangles are 45°-45°-90°
y = 1 + 1 2
Let’s draw the bottom triangle and label the hypotenuse as y
y = 2 2
Let’s apply the Pythagorean Theorem to find the hypotenuse.
y = 2 2
y = 2

11. Right Triangle x x
Can we generalize our findings?

12. In a 45°-45°-90° triangle, the hypotenuse is times as long as a leg
And both legs are the same size. 2
45° s 2 s
45° s

13. . 36 = x 2 36 = x 2 36 2 x If y = x 36 2 = x 2 then 2 y = OR 2 = x y 2
Find the values of the variables. Round your answers to the nearest tenth.
36 = x 2
36 = x 2
45° 36 2 x .
If y = x 36 2
= x 2
then 2 18
y =
45° OR 2 36
= x y 2
( )
y = 25.5
Is this a 45°-45°-90°?
90°-45°=45° s 2
45° 2 36
= x 2 18
= x 2 18
x = OR
x = 25.5

14. . 42 = x 2 42 = x 2 42 2 x If y = x 42 2 = x 2 then 2 y = 2 = x y 2
Find the values of the variables. Give an exact answer.
42 = x 2
42 = x 2
45° 42 2 x .
If y = x 42 2
= x 2
then 2 21
y =
45° 2 42
= x y 2
( )
Is this a 45°-45°-90°?
90°-45°=45° s 2
45° 2 42
= x 2 21
= x 2 21
x =

15. Find the values of the variables. Give the exact answer.
45° 2 x y= y x 2 21
y =
45° 21
Is this a 45°-45°-90°?
90°-45°=45° s 2
45°