1. 7-4: special right triangles
Geometry Chapter 7
7-4: special right triangles

2. Warm-Up Simplify the following. 1.) 10 × 30 2.) 45 5
1.) 10 × )
3.) ) 3 × 27

3. Special Right Triangles
Objective: Students will be able to use the relationships amongst the sides in special right triangles to find side lengths.
Agenda
45°−45°−90° Triangles
30°−60°−90° Triangles
Examples

4. 45°−45°−90° Triangles Definition
A 45°−45°−90° Triangle is an isosceles right Triangle, with 45° as the measures of both the other two angles.
45°
Hypotenuse
Leg

5. 45°−45°−90° Triangles Definition
A 45°−45°−90° Triangle is an isosceles right Triangle, with 45° as the measures of both the other two angles.
Knowledge Connection
Both Legs in this triangle are congruent.
45°
Hypotenuse
Leg

6. 45°−45°−90° Theorem 𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
Theorem 7.8: In a 45°−45°−90° right triangle, the hypotenuse is times as long as a leg.
45° 𝒄 𝒂 𝒃
Hypotenuse
Leg
𝐻𝑦𝑝=𝐿𝑒𝑔 × 2

7. 45°−45°−90° Examples
Find the value of x.

8. 45°−45°−90° Examples
Find the value of x.
Leg
45°
Hypotenuse

9. 45°−45°−90° Examples Find the value of x. 45° Solution: 𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
Leg
Solution:
𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
𝑥=12× 2
𝒙=𝟏𝟐 𝟐
45°
Hypotenuse

10. 45°−45°−90° Examples
Find the value of x.

11. 45°−45°−90° Examples Find the value of x. 45° Solution: 𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
8=x× 2
Leg
Hypotenuse

12. 45°−45°−90° Examples Find the value of x. 45° Solution: 𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
8=x× 2
𝑥= × =
𝒙=𝟒 𝟐
Leg
Hypotenuse

13. 45°−45°−90° Examples
Find the values of x and y.

14. 45°−45°−90° Examples
Find the value of x and y.
Hypotenuse
45°
Leg
Leg

15. 45°−45°−90° Examples Find the value of x and y. 45° For x: 𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
2 6 =x× 2 𝑥=
𝒙=𝟐 𝟑
Hypotenuse
45°
Leg
Leg

16. 45°−45°−90° Examples Find the value of x and y. 45° For x: 𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
Hypotenuse
For x:
𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
2 6 =x× 2 𝑥=
𝒙=𝟐 𝟑
For y:
In a 45°−45°−90° triangle, the Legs have the same length.
Therefore, 𝒚=𝟐 𝟑
45°
Leg
Leg

17. 45°−45°−90° Examples
Find the value of x. 𝟖 𝒙

18. 45°−45°−90° Examples
Find the value of x. 𝟖 𝒙
Hypotenuse
Leg
Leg

19. 45°−45°−90° Examples Find the value of x. 𝒙 𝟖 For x: 𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
𝑥=8× 2
𝒙=𝟖 𝟐 𝟖 𝒙
Hypotenuse
Leg
Leg

20. 30°−60°−90° Triangles Definition
A 30°−60°−90° is a right triangle with 30° and 60° as its other angle measures.
Shorter Leg
Longer Leg
Hypotenuse

21. 30°−60°−90° Triangles Definition
A 30°−60°−90° is a right triangle with 30° and 60° as its other angle measures.
Knowledge Connection
The leg Opposite the 30° angle is called the Shorter Leg.
The Leg Opposite the 60° angle is called the Longer Leg.
Shorter Leg
Longer Leg
Hypotenuse

22. 30°−60°−90° Theorem 𝐻𝑦𝑝=𝑆.𝐿. ×2 𝐿.𝐿. =𝑆.𝐿. × 3
Theorem 7-9: In a 30°−60°−90° right triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as a shorter leg. 𝒄 𝒂 𝒃
Shorter Leg
Longer Leg
Hypotenuse
𝐻𝑦𝑝=𝑆.𝐿. ×2
𝐿.𝐿. =𝑆.𝐿. × 3

23. 30°−60°−90° Examples
Find the values of x and y.

24. 30°−60°−90° Examples Find the values of x and y. Shorter Leg
Longer Leg
Hypotenuse

25. 30°−60°−90° Examples Find the values of x and y. For x: 𝐻𝑦𝑝=𝑆.𝐿. ×2
𝐻𝑦𝑝=𝑆.𝐿. ×2
𝑥=6×2
𝒙=𝟏𝟐
Shorter Leg
Longer Leg
Hypotenuse

26. 30°−60°−90° Examples Find the values of x and y. For x: 𝐻𝑦𝑝=𝑆.𝐿. ×2
𝐻𝑦𝑝=𝑆.𝐿. ×2
𝑥=6×2
𝒙=𝟏𝟐
Shorter Leg
Longer Leg
For y:
𝐿.𝐿. =𝑆.𝐿. × 3
𝑦=6× 3
𝒚=𝟔 𝟑
Hypotenuse

27. 30°−60°−90° Examples
Find the values of x and y. 𝒙 𝒚 𝟐𝟎
𝟔𝟎°

28. 30°−60°−90° Examples Find the values of x and y. 𝒚 𝒙 𝟐𝟎 𝟔𝟎° Longer Leg
Shorter Leg
Hypotenuse

29. 30°−60°−90° Examples Find the values of x and y. 𝒚 𝒙 𝟐𝟎 𝟔𝟎° For x:
𝐻𝑦𝑝=𝑆.𝐿. ×2
20=2x
𝒙=𝟏𝟎
Longer Leg 𝒙 𝒚 𝟐𝟎
𝟔𝟎°
Shorter Leg
Hypotenuse

30. 30°−60°−90° Examples Find the values of x and y. 𝒚 𝒙 𝟐𝟎 𝟔𝟎° For x:
𝐻𝑦𝑝=𝑆.𝐿. ×2
20=2x
𝒙=𝟏𝟎
For y:
𝐿.𝐿. =𝑆.𝐿. × 3
𝑦=10× 3
𝒚=𝟏𝟎 𝟑
Longer Leg 𝒙 𝒚 𝟐𝟎
𝟔𝟎°
Shorter Leg
Hypotenuse

31. 30°−60°−90° Examples
Find the values of x and y.

32. 30°−60°−90° Examples Find the values of x and y. Shorter Leg
Longer Leg
Hypotenuse

33. 30°−60°−90° Examples Find the values of x and y. For x: 𝐿.𝐿. =𝑆.𝐿. × 3
𝐿.𝐿. =𝑆.𝐿. × 3
8=x 3
𝑥= 8 3
𝑥= × = 𝟖 𝟑 𝟑
Shorter Leg
Longer Leg
Hypotenuse

34. 30°−60°−90° Examples Find the values of x and y. For x: 𝐿.𝐿. =𝑆.𝐿. × 3
𝐿.𝐿. =𝑆.𝐿. × 3
8=x 3
𝑥= 8 3
𝑥= ∗ = 𝟖 𝟑 𝟑
For y:
𝐻𝑦𝑝=𝑆.𝐿. ×2
𝑦=𝑥×2
𝑦=2×
𝒚= 𝟏𝟔 𝟑 𝟑
Shorter Leg
Longer Leg
Hypotenuse

35. 30°−60°−90° Examples
Find the values of x and y. 𝟔 𝒙 𝟑

36. 30°−60°−90° Examples Find the values of x and y. 𝟔 𝒙 𝟑 Hypotenuse
Longer Leg
Shorter Leg

37. 30°−60°−90° Examples Find the values of x and y. 𝟔 𝒙 𝟑 For x:
𝐿.𝐿. =𝑆.𝐿. × 3
x=3× 3
𝒙=𝟑 𝟑 𝟔 𝒙 𝟑
Hypotenuse
Longer Leg
Shorter Leg

38. Final Practice: Both Triangles
Find the values of the variables in the given diagram.
For u:
𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
8 2 =u× 2 𝑢=
𝒖=𝟖
For v:
In a 45°−45°−90° triangle, the Legs have the same length.
Therefore, 𝐯=𝟖

39. Final Practice: Both Triangles
Find the values of the variables in the given diagram. 𝒏 𝒎 𝟏𝟎
𝟒𝟓°
For m:
𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
10 =m× 2 𝑚=
𝒎= 𝟓
For n:
In a 45°−45°−90° triangle, the Legs have the same length.
Therefore, 𝐧= 𝟓

40. Final Practice: Both Triangles
Find the values of the variables in the given diagram.
For a:
𝐻𝑦𝑝=𝐿𝑒𝑔 × 2
𝑎=2 2 × 2
𝑎=2(2)
𝒂=𝟒
For b:
In a 45°−45°−90° triangle, the Legs have the same length.
Therefore, 𝐛=𝟐 𝟐

41. Final Practice: Both Triangles
Find the values of the variables in the given diagram.
For u:
𝐻𝑦𝑝=𝑆.𝐿. ×2
u=2×2
𝒖=𝟒
For v:
𝐿.𝐿. =𝑆.𝐿. × 3
𝑦=2× 3
𝒚=𝟐 𝟑

42. Final Practice: Both Triangles
Find the values of the variables in the given diagram.
For y:
𝐿.𝐿. =𝑆.𝐿. × 3
𝑦=4 5 × 3
𝒚=𝟒 𝟏𝟓
For y:
𝐻𝑦𝑝=𝑆.𝐿. ×2
8 5 =2y
𝒚=𝟒 𝟓

43. Final Practice: Both Triangles
Find the values of the variables in the given diagram.
For b:
𝐿.𝐿. =𝑆.𝐿. × 3
11 3 =𝑏× 3 𝑏=
𝒃=𝟏𝟏
For a:
𝐻𝑦𝑝=𝑆.𝐿. ×2
a=11×2
𝒂=𝟐𝟐