Geometric Mean in Triangles Calculator
Altitude to hypotenuse relationships
Input Method
Geometric Mean Theorems
- • Altitude: h² = x · y
- • Leg a: a² = x · c
- • Leg b: b² = y · c
Right Triangle with Altitude
Understanding Geometric Mean in Right Triangles
When an altitude is drawn from the right angle to the hypotenuse of a right triangle, it creates three similar triangles. This similarity leads to beautiful geometric mean relationships.
The Three Geometric Mean Theorems
Altitude-on-Hypotenuse Theorem
The altitude to the hypotenuse is the geometric mean of the two segments:
h² = x · y → h = √(xy)
Leg-Hypotenuse Theorem
Each leg is the geometric mean of the hypotenuse and its adjacent segment:
a² = x · c and b² = y · c
Why These Relationships Work
Drawing the altitude from the right angle creates three similar triangles:
- △ABC ~ △ACD ~ △CBD (original and two smaller triangles)
- Corresponding sides of similar triangles are proportional
- Setting up proportions leads to the geometric mean relationships
Derivation from Similar Triangles
From △ACD ~ △CBD:
x/h = h/y → h² = xy
From △ABC ~ △ACD:
c/a = a/x → a² = xc
From △ABC ~ △CBD:
c/b = b/y → b² = yc
Frequently Asked Questions
What is the geometric mean?
The geometric mean of two numbers a and b is √(ab). It's the number x such that a/x = x/b, meaning x² = ab.
Why is the altitude a geometric mean?
The altitude creates two similar triangles. The proportions between corresponding sides give h/x = y/h, which rearranges to h² = xy.
How do I find the altitude if I know the legs?
Use h = ab/c, where a and b are the legs and c = √(a² + b²) is the hypotenuse. This is equivalent to Area = (1/2)ab = (1/2)ch.
Can I verify the Pythagorean theorem using these relationships?
Yes! Since a² = xc and b² = yc, we have a² + b² = xc + yc = (x+y)c = c·c = c². This proves a² + b² = c².
What's the relationship between segments x and y?
The segments always add up to the hypotenuse: x + y = c. Also, the ratio x:y equals a²:b², the ratio of the squared legs.
Are the three small triangles always similar?
Yes! All three triangles (the original and two created by the altitude) are similar because they share the same angles. This is the foundation of all geometric mean relationships.