9. Geometric Mean Theorem Calculator (right Triangle Altitude)
In a right triangle, the altitude from the right angle to the hypotenuse creates two smaller right triangles that are
similar to the original. This similarity leads to a powerful set of relations called the
Geometric Mean Theorem.
Setup
Let \(ABC\) be a right triangle with right angle at \(C\). The hypotenuse is \(AB=c\).
Drop the altitude \(CD=h\) to the hypotenuse, meeting \(AB\) at \(D\).
Then the hypotenuse is split into two segments:
\[
\begin{aligned}
AD &= p \\
DB &= q \\
c &= p+q
\end{aligned}
\]
The legs are \(AC=a\) and \(BC=b\).
Main formulas
1) Altitude is the geometric mean of the segments:
\[
\begin{aligned}
h^2 &= pq \\
h &= \sqrt{pq}
\end{aligned}
\]
2) Each leg is the geometric mean of the hypotenuse and its adjacent segment:
\[
\begin{aligned}
a^2 &= cp \\
b^2 &= cq
\end{aligned}
\]
3) Useful combined identity:
\[
\begin{aligned}
h &= \frac{ab}{c}
\end{aligned}
\]
How the calculator solves problems
- From legs \(a,b\): compute \(c=\sqrt{a^2+b^2}\), then \(p=a^2/c\), \(q=b^2/c\), and \(h=ab/c\).
- From segments \(p,q\): compute \(c=p+q\), \(h=\sqrt{pq}\), \(a=\sqrt{cp}\), \(b=\sqrt{cq}\).
- From \(c\) and \(h\): solve \(p+q=c\) and \(pq=h^2\), which forms a quadratic and yields two swapped solutions.
- Verify mode: checks consistency of the relations and reports PASS/FAIL using relative error.
Worked example (5–12–13)
Given legs \(a=5\) and \(b=12\), the hypotenuse is \(c=13\).
The altitude to the hypotenuse is:
\[
\begin{aligned}
h &= \frac{ab}{c}
= \frac{5\cdot 12}{13}
= \frac{60}{13}
\approx 4.615
\end{aligned}
\]
The hypotenuse segments are:
\[
\begin{aligned}
p &= \frac{a^2}{c} = \frac{25}{13} \approx 1.923 \\
q &= \frac{b^2}{c} = \frac{144}{13} \approx 11.08
\end{aligned}
\]
Check: \(h^2 = pq\) → \(\left(\frac{60}{13}\right)^2 = \frac{25}{13}\cdot \frac{144}{13}\) ✔
Common mistakes
- Mixing up which segment is \(p\) vs \(q\) (they can swap; the geometry stays the same).
- Using non-right-triangle inputs (the theorem assumes a right triangle).
- Entering values that make \(c^2 - 4h^2 < 0\) in the \((c,h)\) mode (no real segment split exists).
