In a right triangle, drawing the altitude from the right angle to the hypotenuse creates a powerful set of relationships. The altitude splits the original triangle into two smaller right triangles, and all three triangles are similar. This makes it possible to find missing lengths using proportions instead of only the Pythagorean theorem.
These relationships are especially useful in geometry proofs, construction problems, and coordinate geometry.
Key Facts
- If CD is the altitude to hypotenuse AB, then AD = p, DB = q, and AB = p + q.
- The three triangles are similar: △ABC ∼ △ACD ∼ △CBD.
- Altitude geometric mean formula: CD^2 = AD · DB, so CD = sqrt(pq).
- Leg AC geometric mean formula: AC^2 = AB · AD, so AC = sqrt((p + q)p).
- Leg BC geometric mean formula: BC^2 = AB · DB, so BC = sqrt((p + q)q).
- The Pythagorean theorem still applies to the original triangle: AC^2 + BC^2 = AB^2.
Vocabulary
- Geometric mean
- The geometric mean of two positive numbers a and b is sqrt(ab), or the positive number x such that x^2 = ab.
- Altitude
- An altitude is a perpendicular segment from a vertex of a triangle to the opposite side or the line containing it.
- Hypotenuse
- The hypotenuse is the side opposite the right angle in a right triangle and is always the longest side.
- Similar triangles
- Similar triangles have equal corresponding angles and proportional corresponding side lengths.
- Projection
- A projection is the segment on the hypotenuse formed by dropping a perpendicular from the right angle to the hypotenuse.
Common Mistakes to Avoid
- Using CD = p · q instead of CD^2 = p · q is wrong because the altitude is the geometric mean, not the product of the two hypotenuse segments.
- Mixing up AD and DB in the leg formulas is wrong because AC pairs with AD and BC pairs with DB based on their positions in the similar triangles.
- Forgetting that AB = p + q is wrong because the full hypotenuse is the sum of the two smaller segments created by point D.
- Assuming the two smaller triangles are congruent is wrong because they are only similar unless p and q are equal.
Practice Questions
- 1 In right triangle △ABC, ∠C = 90°, CD is perpendicular to AB, AD = 9, and DB = 16. Find CD, AB, AC, and BC.
- 2 In right triangle △ABC, CD is the altitude to hypotenuse AB. If AB = 25 and AD = 7, find DB, AC, and BC.
- 3 Explain why drawing altitude CD to hypotenuse AB creates three similar right triangles, and describe how that similarity leads to the formula CD^2 = AD · DB.