1. Geometric Mean Pythagorean Theorem Special Right Triangles

2. Geometric Mean
When the means of a proportion are the same number, that number is called the geometric mean of extremes.

3. Geometric Mean
The geometric mean between two numbers is the positive square root of their product.

4. Examples
Find the geometric mean between the pair of numbers.
5 and 45

5. Examples Find the geometric mean between the pair of numbers. 5 and 45

6. Examples Find the geometric mean between the pair of numbers.
12 and 15

7. Examples Find the geometric mean between the pair of numbers.
12 and 15
x2 = 12 * 15 = 180
x = √180 = 6√5

8. Geometric Means in Triangles
In a right triangle, an altitude drawn from the vertex of the right triangle to the hypotenuse forms two additional right triangles. These three right triangles share a special relationship.

9. Geometric Means in Triangles
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.

10. Geometric Mean (Altitude) Theorem
The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these two segments.

11. Geometric Mean (Leg) Theorem
The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

12. Geometric Mean (Leg) Theorem

13. Examples
Find x, y, and z.

14. Examples Find x, y, and z. z = √8*25 = √200 z = 10√2

15. Examples
Find x, y, and z.

16. Examples Find x, y, and z. 12 = √(9*x) 144 = 9x 16 = x y = √(16*25)

17. Pythagorean Theorem
In a right triangle, the sum of the squares of the lengths of the sides is equal to the square of the length of the hypotenuse.

18. Pythagorean Triple
A Pythagorean triple is a set of three nonzero whole numbers a, b, and c, such that
a2 + b2 = c2.

19. Examples
Use a Pythagorean triple to find x. Explain your reasoning.

20. Examples Use a Pythagorean triple to find x. Explain your reasoning.
20 = 5*4
48 = 12*4
x = 13*4 = 52

21. Pythagorean Inequality Theorem
If the square of the length of the hypotenuse is less than the sum of the squares of the lengths of the other two sides, then the triangle is an acute triangle.

22. Pythagorean Inequality Theorem
If the square of the length of the hypotenuse is greater than the sum of the squares of the lengths of the other two sides, then the triangle is an obtuse triangle.

23. Triangle Theorem
In a triangle, the legs l are congruent and the length of the hypotenuse h is √2 times the length of a leg.

24. Examples
Find x.

25. Examples
Find x.
x = 5√2

26. Examples
Find x.

27. Examples
Find x.
18 = x√2
x = 18/√2
x = 9√2

28. Examples
Find x.

29. Examples
Find x.
x = 7√2 *√2
x = 7*2
x = 14

30. Triangle Theorem
In a triangle, the length of the hypotenuse is 2 times the length of the shorter leg and the longer leg is √3 times the length of the shorter leg.

31. Examples
Find x and y.

32. Examples Find x and y. 15 = x√3 x = 15/√3 x = 5√3 y = 2x

33. Examples
Find x and y.

34. Examples
Find x and y.
x = 12/√3
x = 4√3
y = 2*x
y = 2 * 4√3
y = 8√3

35. Examples
Find x and y.
x = .5 * 10
x = 5
y = x * √3
y = 5√3