4.4 Right-Triangle Altitude & Geometric Mean

The Altitude to the Hypotenuse

When you draw the altitude from the right angle to the hypotenuse of a right triangle, something remarkable happens: it splits the original triangle into two smaller right triangles, and all three triangles are similar to one another. This result, which flows directly from the AA criterion, powers the geometric mean relationships and even gives a similarity proof of the Pythagorean Theorem (Next Generation standard G-SRT.B.4).

Why are they similar? In right triangle ABC with the right angle at C, drop altitude CD to hypotenuse AB. Each smaller triangle shares an acute angle with the big triangle and has its own right angle, so by AA every triangle here is similar:

triangle ACD ~ triangle CBD ~ triangle ABC

The hard part on the exam is matching corresponding parts across these three overlapping triangles. That is exactly where the geometric mean shortcuts save time.

The Geometric Mean

The geometric mean of two positive numbers a and b is the value x with a/x = x/b, so x = sqrt(a * b). For example, the geometric mean of 4 and 9 is sqrt(36) = 6. Two clean rules come from the three similar triangles above.

Rule 1: The Altitude Rule

The altitude to the hypotenuse is the geometric mean of the two hypotenuse segments it creates. If the altitude h divides the hypotenuse into pieces p and q:

h = sqrt(p * q), equivalently h^2 = p * q

Worked example. The altitude to the hypotenuse divides it into segments of length 9 and 16. Then h^2 = 9 * 16 = 144, so h = 12.

Rule 2: The Leg Rule

Each leg is the geometric mean of the whole hypotenuse and the hypotenuse segment adjacent to that leg. If a leg L sits next to the hypotenuse segment p, and the full hypotenuse is c:

L = sqrt(c * p), equivalently L^2 = c * p

Worked example. A leg is adjacent to a hypotenuse segment of length 9, and the whole hypotenuse is 25. Then L^2 = 25 * 9 = 225, so L = 15.

A quick way to keep the rules apart: the altitude uses the two little pieces (p and q); a leg uses the whole hypotenuse and the one piece it touches (c and p).

Worked Examples Solving for Segments

Example A (find a segment from the altitude). The altitude to the hypotenuse has length 12, and one hypotenuse segment is 16. Using h^2 = p * q, we get 12^2 = 16 * q, so 144 = 16q and q = 9. The full hypotenuse is 16 + 9 = 25.

Example B (find a leg). A right triangle has hypotenuse segments 4 and 9 next to the two legs. The full hypotenuse is 13. The leg next to the 4-segment is sqrt(13 * 4) = sqrt(52) = 2sqrt(13); the leg next to the 9-segment is sqrt(13 * 9) = sqrt(117) = 3sqrt(13). As a check, the altitude is sqrt(4 * 9) = 6, and 6 is the geometric mean of 4 and 9.

Example C (mixed). The two hypotenuse segments are 3 and 12. The altitude is sqrt(3 * 12) = sqrt(36) = 6. The hypotenuse is 15, so the legs are sqrt(15 * 3) = sqrt(45) = 3sqrt(5) and sqrt(15 * 12) = sqrt(180) = 6sqrt(5). Notice 6 (altitude) times 15 (hypotenuse) equals 90, which also equals the product of the legs, another handy relationship these similar triangles produce.

Watch the Traps

• Do not add the segments and then take the square root. The altitude uses the product p * q, not the sum. With segments 9 and 16, h = sqrt(144) = 12, not sqrt(25) = 5.
• Match each leg to the segment it touches. Pairing a leg with the far segment produces a wrong answer.
• Leave answers in simplest radical form unless the question says to round, so sqrt(52) becomes 2*sqrt(13). The Regents reference sheet and calculator both help, but exact form is often expected on constructed-response work.

A Similarity Proof of the Pythagorean Theorem

The leg rule quietly proves the Pythagorean Theorem. With hypotenuse c split into segments p and q, the two legs satisfy a^2 = cp and b^2 = cq. Adding these: a^2 + b^2 = cp + cq = c(p + q) = c*c = c^2, because p + q is the whole hypotenuse c. This is the similarity-based derivation the Next Generation standards highlight, and it shows the geometric-mean relationships are structural facts about right triangles, not isolated tricks.

One More Worked Example

A right triangle has legs 6 and 8 and hypotenuse 10. Find the altitude to the hypotenuse two ways. By area: the area is (1/2)(6)(8) = 24, and also (1/2)(10)(h), so 5h = 24 and h = 4.8.

Check with geometric means: the foot of the altitude splits the hypotenuse into p = 6^2/10 = 3.6 and q = 8^2/10 = 6.4, and sqrt(3.6 * 6.4) = sqrt(23.04) = 4.8. Two independent methods agree, exactly the kind of self-check that catches errors under time pressure.