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Geometry Similarity

Geometric mean (altitude to hypotenuse of a right triangle)

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The Geometric Mean in Right Triangles

Common Core Geometry • Standard G-SRT.5 • Similarity

The Geometric Mean in Right Triangles is a topic in Similarity in the 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.

Common Core Geometry › Similarity › The Geometric Mean in Right Triangles  —  Standard G-SRT.5

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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.
Similarity is the source. The proportions from the similar triangles are exactly the geometric-mean relationships.
Geometric mean in a right triangle The altitude to the hypotenuse of a right triangle is the geometric mean of the two hypotenuse segments. p q h altitude h = geometric mean of p and q
The altitude \(h\) is the geometric mean of the segments \(p\) and \(q\).
Geometric mean relationships Geometric mean relationships Geometric mean relationships altitude: h = √(p · q) leg: a = √(hyp · adjacent segment)
The altitude and leg geometric-mean formulas.

The geometric-mean relationships:

\[h=\sqrt{pq},\qquad \text{leg}=\sqrt{(\text{hypotenuse})(\text{adjacent segment})}\]
the altitude is the square root of p times q; a leg is the square root of the hypotenuse times its adjacent segment
Geometric mean of \(p,q\) is \(\sqrt{pq}\) — not the average \(\dfrac{p+q}{2}\).

How to use the geometric mean

  1. Identify whether you need the altitude or a leg.
  2. Altitude: \(h=\sqrt{pq}\) from the two segments.
  3. Leg: \(\sqrt{\text{hypotenuse}\times\text{adjacent segment}}\).
  4. Simplify the radical.
Example 1 — Geometric mean of two numbers
Find the geometric mean of \(4\) and \(9\).
Solution

The geometric mean of \(p\) and \(q\) is \(\sqrt{pq}\).

\(\sqrt{4\cdot 9}\)\(=\)\(\sqrt{36}=6\)
the geometric mean is 6
Example 2 — Altitude to the hypotenuse
The altitude to the hypotenuse splits it into \(p=3\) and \(q=12\). Find the altitude \(h\).
Solution

The altitude is the geometric mean of the two segments.

\(h\)\(=\)\(\sqrt{3\cdot 12}\)
\(=\)\(\sqrt{36}=6\)
the altitude is 6
Example 3 — A leg as a geometric mean
A hypotenuse is \(16\); the segment adjacent to leg \(a\) is \(4\). Find \(a\).
Solution

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 leg is 8
Example 4 — Why the geometric mean appears
Why does the altitude create a geometric mean?
Solution

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 \(h=\sqrt{pq}\).

the similar triangles give the proportion p over h equals h over q

Common pitfalls

Geometric mean is \(\sqrt{pq}\), not the ordinary average.
Match a leg with its adjacent segment, the one it touches on the hypotenuse.
The altitude uses both segments; a leg uses the whole hypotenuse. Don't mix them up.

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.

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