Geometric Mean

Definition

The geometric mean of two positive numbers a and b is x = √(ab). It is the positive number x such that a/x = x/b.

Geometric Mean in Right Triangles

When the altitude is drawn from the right Angle to the hypotenuse of a right Triangle, it creates two smaller triangles that are similar to each other and to the original Triangle.

Three Key Relationships

In right △ABC with altitude CD to hypotenuse AB (where D is the foot):

• Altitude rule: CD = √(AD × DB) — the altitude is the geometric mean of the two segments of the hypotenuse.
• Leg rule: AC = √(AB × AD) — each leg is the geometric mean of the hypotenuse and the adjacent segment.
• Leg rule: BC = √(AB × DB)