Altitude Theorem and Leg Theorem
The altitude theorem and the leg theorem describe how the altitude to the hypotenuse of a right triangle divides it into two smaller right triangles and gives simple formulas for lengths. In brief: if the altitude h splits the hypotenuse c into segments p and q, then h2 = p · q, a2 = c · p and b2 = c · q. For p = 4 and q = 9, the altitude is h = √36 = 6.
These two results, also called the geometric mean theorems or Euclid’s theorems, follow from the similarity of triangles. Together with the Pythagorean theorem they allow you to find all the lengths in a right triangle from only two pieces of data. This guide explains the setup, the formulas, a proof, worked examples and where they are used.
What is the altitude to the hypotenuse?
Take a right triangle ABC with the right angle at C, legs a = BC and b = AC and hypotenuse c = AB. The altitude from C to the hypotenuse is the perpendicular segment from C to the line AB, with foot D and length h. The point D divides the hypotenuse into two parts: p = BD, next to leg a, and q = AD, next to leg b. So c = p + q.
The two small triangles ACD and CBD are right triangles.
What does the altitude theorem say?
The square of the altitude equals the product of the two segments of the hypotenuse: h2 = p · q. In other words, h is the geometric mean of p and q: h = √(pq). It can be used to construct a square equal in area to a given rectangle, a classical use in Greek geometry.
What does the leg theorem say?
The square of a leg equals the product of the hypotenuse and the segment adjacent to that leg: a2 = c · p and b2 = c · q. Adding the two equations gives a2 + b2 = c(p + q) = c2, which is the Pythagorean theorem. So the leg theorem gives another proof of it.
Why do the theorems work?
The altitude creates three similar triangles: ABC, CBD and ACD. The triangle CBD shares the angle B with ABC, and both have a right angle, so they are similar. Similarly ACD is similar to ABC. From CBD ~ ABC: BD/BC = BC/AB, which is p/a = a/c, so a2 = cp. From ACD ~ ABC: AD/AC = AC/AB gives q/b = b/c, so b2 = cq. From ACD ~ CBD: AD/CD = CD/BD gives q/h = h/p, so h2 = pq.
The key idea is similar triangles.
Worked examples
Example 1: altitude from the segments
The altitude splits the hypotenuse into 4 and 9. Then h = √(4 · 9) = 6. The hypotenuse is 13. The legs are a = √(13 · 4) = √52 = 2√13 ≈ 7.21 and b = √(13 · 9) = √117 = 3√13 ≈ 10.82. Check with Pythagoras: 52 + 117 = 169 = 132 ✓.
Example 2: from the hypotenuse and a leg
A right triangle has hypotenuse 25 and leg a = 15. Then p = a2/c = 225/25 = 9, and q = 25 − 9 = 16. The altitude is h = √(9 · 16) = 12. The other leg is b = √(25 · 16) = 20, giving the 15-20-25 triangle, a multiple of 3-4-5.
Example 3: from the legs
Legs 6 and 8. The hypotenuse is 10. The altitude can also be computed from the area: (1/2) · 6 · 8 = 24 = (1/2) · 10 · h, so h = 4.8. Check: p = 36/10 = 3.6, q = 6.4 and pq = 23.04 = 4.82 ✓.
Formula summary
| Theorem | Formula | Example with p = 4, q = 9 |
|---|---|---|
| Altitude theorem | h2 = p · q | h = 6 |
| Leg theorem for a | a2 = c · p | a = √52 ≈ 7.21 |
| Leg theorem for b | b2 = c · q | b = √117 ≈ 10.82 |
| Hypotenuse | c = p + q | c = 13 |
| Altitude from area | h = ab/c | h = 7.21 · 10.82/13 ≈ 6 |
Where do the theorems come from?
Both theorems are classical and are connected with Euclid’s Elements, which includes the construction of a mean proportional using a right triangle and the altitude. Their names vary among countries: in some, the leg theorem is called Euclid’s theorem and the altitude theorem is the height theorem. The statement as a pair of algebraic formulas is a modern way of writing the classical proportions.
Where are they used today?
The geometric mean construction is used in drafting to build a square of a given area with a compass and a straightedge. In practice, the theorems help to find heights in a right triangle that cannot be measured directly, and they appear in problems about ramps and roof frames. They are also a standard tool in competition geometry.
Which mistakes are common?
Mixing up p and q
The leg a goes with the segment p adjacent to it. Draw a diagram and label the segments carefully.
Using the theorems for non-right triangles
They need a right angle at C and the altitude to the hypotenuse.
Forgetting the square root
h2 = pq gives h = √(pq), not pq.
Confusing the altitude with a median
The altitude is perpendicular to the hypotenuse. The median goes to the midpoint and has length c/2 in a right triangle.