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!