Geometric mean (altitude to hypotenuse of a right triangle)
The Geometric Mean in Right Triangles
The Geometric Mean in Right Triangles is a topic in Similarity in the California Common Core State Standards. It is aligned to Standard G-SRT.5, which requires students to use similarity relationships in right triangles to solve problems, including geometric-mean relationships.
The altitude to the hypotenuse of a right triangle is the geometric mean of the two segments it cuts on the hypotenuse.
Theory
The geometric mean of two positive numbers \(p\) and \(q\) is \(\sqrt{pq}\).
When the altitude to the hypotenuse is drawn in a right triangle, it splits the triangle into two smaller triangles, each similar to the original. This produces two geometric-mean relationships:
- the altitude is the geometric mean of the two hypotenuse segments: \(h=\sqrt{pq}\);
- each leg is the geometric mean of the whole hypotenuse and the segment adjacent to it.
The geometric-mean relationships:
How to use the geometric mean
- Identify whether you need the altitude or a leg.
- Altitude: \(h=\sqrt{pq}\) from the two segments.
- Leg: \(\sqrt{\text{hypotenuse}\times\text{adjacent segment}}\).
- Simplify the radical.
The geometric mean of \(p\) and \(q\) is \(\sqrt{pq}\).
| \(\sqrt{4\cdot 9}\) | \(=\) | \(\sqrt{36}=6\) |
The altitude is the geometric mean of the two segments.
| \(h\) | \(=\) | \(\sqrt{3\cdot 12}\) |
| \(=\) | \(\sqrt{36}=6\) |
A leg is the geometric mean of the whole hypotenuse and the segment adjacent to it.
| \(a\) | \(=\) | \(\sqrt{16\cdot 4}\) |
| \(=\) | \(\sqrt{64}=8\) |
The altitude splits the right triangle into two smaller triangles similar to each other and to the whole; matching proportional sides gives \(\dfrac{p}{h}=\dfrac{h}{q}\), so \(h=\sqrt{pq}\).
Common pitfalls
Frequently asked questions
What is the geometric mean of two numbers?
The square root of their product: the geometric mean of \(p\) and \(q\) is \(\sqrt{pq}\).
What is the altitude-on-hypotenuse relationship?
The altitude to the hypotenuse is the geometric mean of the two segments it creates: \(h=\sqrt{pq}\).
How is a leg a geometric mean?
Each leg is the geometric mean of the whole hypotenuse and the hypotenuse segment adjacent to that leg.
Why do these geometric means appear?
Because the altitude creates two triangles similar to the original; their proportional sides give the geometric-mean equations.