Geometric Mean and Similar Triangles

Geometric Mean

Positive number x such that a/x = x/b

Theorem 8.1

Altitude to hypotenuse creates similar triangles

Corollary 8.1.1

Altitude length is geometric mean of hypotenuse segments

Corollary 8.1.2

Leg is geometric mean between hypotenuse and adjacent segment

Radical in Simplest Form

Radical rules: no perfect square, no fractions

Pythagorean Theorem

Hypotenuse squared equals sum of legs squared

Converse of Pythagorean Theorem

Sum of squares of sides determines right triangle

Theorem 8.4

Longest side squared relation to other sides

Theorem 8.5

Longest side squared relation to other sides for obtuse triangle

45-45-90 Theorem

Hypotenuse is √2 times a leg in 45-45-90 triangle

30-60-90 Theorem

Hypotenuse, legs ratio in 30-60-90 triangle

Tangent Ratio

Ratio of opposite leg to adjacent leg

Sine Ratio

Ratio of opposite leg to hypotenuse

Cosine Ratio

Ratio of adjacent leg to hypotenuse

Angle of Depression/Elevation

Angle between horizontal and line of sight