Pythagorean Theorem and Special Right Triangles April 28, 2008

Similarity What makes two polygons similar?

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.

Working with geometric mean Try these

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.

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.

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.

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

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.

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

30-60-90 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.

Try it out… Find the missing values.

Some more examples Find the missing values.

A harder example

Another

Yet another

Last one!