Similarity in right triangles and the geometric mean
How the altitude to the hypotenuse splits a right triangle into similar triangles, the geometric mean, and the altitude and leg rules for solving right triangles.
Three similar triangles
Take right triangle \(\triangle ABC\) with the right angle at \(C\), and draw the altitude \(\overline{CD}\) to the hypotenuse \(\overline{AB}\). The altitude splits the triangle into two smaller right triangles, \(\triangle ACD\) and \(\triangle CBD\).
Why: \(\triangle ACD\) and \(\triangle ABC\) share \(\angle A\) and each has a right angle, so they're similar by AA. The same argument works for \(\triangle CBD\) and \(\triangle ABC\), which share \(\angle B\).
The geometric mean
The geometric mean of \(8\) and \(18\) is \(\sqrt{144} = 12\). Check: \(\frac{8}{12} = \frac{12}{18} = \frac{2}{3}\).
Two rules from the similar triangles
The altitude splits the hypotenuse into two segments. Writing proportions between the similar triangles gives two rules.
Solving right triangles
Practice
Quick check
The altitude to the hypotenuse divides it into segments of \(2\) and \(8\). How long is the altitude?
\(\sqrt{2 \cdot 8} = \sqrt{16} = 4\).
Quick check
Which equation is the leg rule for a leg \(a\), hypotenuse \(c\) and adjacent segment \(s\)?
Each leg is the geometric mean of the hypotenuse and its adjacent segment.
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