1. 7.3 Special Right Triangles

2. Objectives Use properties of 45° - 45° - 90° triangles

3. Side Lengths of Special Right ∆s
Right triangles whose angle measures are 45° - 45° - 90° or 30° - 60° - 90° are called special right triangles. The theorems that describe the relationships between the side lengths of each of these special right triangles are as follows:

4. 45° - 45° - 90°∆
Theorem 7.6
In a 45°- 45°- 90° triangle, the length of the hypotenuse is √2 times the length of a leg.
hypotenuse = √2 • leg
45 °
x√2
45 °

5. Example 1:
WALLPAPER TILING The wallpaper in the figure can be divided into four equal square quadrants so that each square contains 8 triangles. What is the area of one of the squares if the hypotenuse of each 45°- 45°- 90° triangle measures millimeters?

6. Example 1:
The length of the hypotenuse of one 45°- 45°- 90° triangle is millimeters. The length of the hypotenuse is times as long as a leg. So, the length of each leg is 7 millimeters.
The area of one of these triangles is
or 24.5 millimeters.
Answer: Since there are 8 of these triangles in one square quadrant, the area of one of these squares is 8(24.5) or 196 mm2.

7. Your Turn:
WALLPAPER TILING If each 45°- 45°- 90° triangle in the figure has a hypotenuse of millimeters, what is the perimeter of the entire square?
Answer: 80 mm

8. Example 2:
Find a.
The length of the hypotenuse of a 45°- 45°- 90° triangle is times as long as a leg of the triangle.

9. Example 2: Divide each side by Rationalize the denominator. Multiply.
Answer:

10. Your Turn:
Find b.
Answer:

11. 30° - 60° - 90°∆
Be sure you realize the shorter leg is opposite the 30° & the longer leg is opposite the 60°.
Theorem 7.7
In a 30°- 60°- 90° triangle, the length of the hypotenuse is twice as long as the shorter leg, and the length of the longer leg is √3 times as long as the shorter leg.
60 °
30 °
x√3
Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg

12. Example 3:
Find QR.

13. Example 3:
is the longer leg, is the shorter leg, and is the hypotenuse.
Multiply each side by 2.
Answer:

14. Your Turn:
Find BC.
Answer: BC = 8 in.

15. Example 4:
COORDINATE GEOMETRY is a 30°-60°-90° triangle with right angle X and as the longer leg. Graph points X(-2, 7) and Y(-7, 7), and locate point W in Quadrant III.

16. Example 4:
Graph X and Y lies on a horizontal gridline of the coordinate plane. Since will be perpendicular to it lies on a vertical gridline. Find the length of

17. Example 4:
is the shorter leg is the longer leg. So, Use XY to find WX.
Point W has the same x-coordinate as X. W is located units below X.
Answer: The coordinates of W are or about

18. Your Turn:
COORDINATE GEOMETRY is at 30°-60°-90° triangle with right angle R and as the longer leg. Graph points T(3, 3) and R(3, 6) and locate point S in Quadrant III.
Answer: The coordinates of S are or about

19. Assignment Pre-AP Geometry: Pg. 360 #12 – 28, 36, 38
Geometry: Pg. 360 #12 – 24 and 36