1. Pythagorean Theorem and Special Right Triangles
April 28, 2008

2. Similarity
What makes two polygons similar?

3. Geometric Mean
For any two positive numbers a and b, the geometric mean, of a and b is the positive number x such that
Find the geometric mean of 4 and 8.

4. Working with geometric mean
Try these

5. Right Triangles
Theorem: The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.

6. Corollary 1
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.

7. Corollary 2
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 the leg.

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

9. Pythagorean Triples
Three integers (like 5,12, and 13) that satisfy the conditions of the Pythagorean Theorem are called Pythagorean Triples.
If the three integers are relatively prime (meaning they have no common factors) then the three integers are know and Primitive Pythagorean Triples.

10. Triangles
45°-45°-90° Theorem In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as a leg.

11. Triangles
30°-60°-90° Theorem 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.

12. Try it out…
Find the missing values.

13. Some more examples
Find the missing values.

14. A harder example

15. Another

16. Yet another

17. Last one!