1. CHAPTER 8
Right Triangles

2. Similarity in Right Triangles
SECTION 8-1
Similarity in Right Triangles

3. Geometric Mean
If a, b, and x are positive numbers and a/x = x/b, then x is called the geometric mean between a and b and x = √ab

4. THEOREM 8 -1
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other

5. Corollary 1
When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse.

6. Corollary 2
When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

7. The Pythagorean Theorem
SECTION 8-2
The Pythagorean Theorem

8. THEOREM
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

9. The Converse of the Pythagorean Theorem
SECTION 8-3
The Converse of the Pythagorean Theorem

10. Triangle
In order to have a triangle, the sum of any two sides must be greater than the third side.

11. THEOREM
If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

12. Some Common Right Triangle Lengths
3,4,5 5,12,13
6,8,10 10,24,26
9,12,15 8,15,17
12,16,20 7,24,25
15,20,25

13. If c2 = a2 + b2, then mC = 90°, and ∆ABC is right
THEOREM 8 - 3
If c2 = a2 + b2, then mC = 90°, and ∆ABC is right

14. If c2 < a2 + b2, then mC < 90°, and ∆ABC is acute
THEOREM 8 - 4
If c2 < a2 + b2, then mC < 90°, and ∆ABC is acute

15. If c2 > a2 + b2, then mC > 90°, and ∆ABC is obtuse
THEOREM 8 - 5
If c2 > a2 + b2, then mC > 90°, and ∆ABC is obtuse

16. Special Right Triangles
SECTION 8-4
Special Right Triangles

17. THEOREM 8 - 6
In a 45°- 45°- 90° triangle, the hypotenuse is √2 times as long as a leg.

18. THEOREM 8 - 7
In a 30°- 60°- 90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.

19. The Tangent and Cotangent Ratios
SECTION 8-5
The Tangent and Cotangent Ratios

20. TRIGONOMETRIC RATIOS
For any right triangle, there are six trigonometric ratios of the lengths of the sides of a triangle.

21. Ratio: length of side opposite A to the length of side adjacent to A
Tangent A
Ratio: length of side opposite A to the length of side adjacent to A

22. Ratio: length of side adjacent A to the length of side opposite to A
Cotangent A
Ratio: length of side adjacent A to the length of side opposite to A
= 1/(tan A)

23. Examples
Find tan M N 13 5 M P 12

24. The Sine, Cosecant, Cosine, and Secant Ratios
SECTION 8-6
The Sine, Cosecant, Cosine, and Secant Ratios

25. Ratio: length of side opposite A to length of hypotenuse.
Sine A
Ratio: length of side opposite A to length of hypotenuse.

26. Examples
Find sin M N 13 5 M P 12

27. Examples
Find sin N N 13 5 M P 12

28. Ratio: length of hypotenuse of A to length of side opposite A.
Cosecant A
Ratio: length of hypotenuse of A to length of side opposite A.
= 1/(sin A)

29. Ratio: Length of side adjacent A to the length of hypotenuse.
Cosine A
Ratio: Length of side adjacent A to the length of hypotenuse.

30. Examples
Find cos M N 13 5 M P 12

31. Ratio: length of hypotenuse of A to length of side adjacent to A.
Secant A
Ratio: length of hypotenuse of A to length of side adjacent to A.
= 1/(cos A)

32. Applications of Right Triangle Trigonometry
SECTION 8-7
Applications of Right Triangle Trigonometry

33. The angle between the horizontal and the line of sight
Angle of Depression
The angle between the horizontal and the line of sight

34. The angle above the horizontal and the line of sight
Angle of Elevation
The angle above the horizontal and the line of sight

35. END