AMC 10 · 2002 · #13

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
area-trianglespythagorean-theorem identify-subproblems ↑ Prerequisites: area-trianglespythagorean-theorem
📏 Medium solution 💡 2 insights
Problem
A triangle has sides of length 15, 20, and 25. Its three altitudes are the perpendicular heights drawn from each vertex to the side across from it; find the length of the shortest altitude.

Pick an answer.

(A)
6
(B)
12
(C)
12.5
(D)
13
(E)
15

AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Extreme Principle

The area of a triangle is 1/2×base×height, and that area is fixed no matter which side you call the base. So for each side s with altitude h, the product s × h always equals twice the area — a constant. Tool #14 (Extreme Principle) reads that directly: if s × h is fixed, then to make h as small as possible you must draw it to the largest side s. So the shortest altitude is the one dropped to the longest side. First find the area (easy, because #5 spots that 15,20,25 is a right triangle), then divide twice the area by the longest side.

1STEP 1

Spot the right triangle

Since 152+202=625=25215^2+20^2=625=25^2, it is a right triangle with legs 15 and 20 and hypotenuse 25 — the 3-4-5 triple times 5.

15²+20² = 225+400 = 625 = 25² → right triangle, legs 15,20, hypotenuse 25
2STEP 2

Find the area

The legs are perpendicular, so they act as base and height: 12×15×20\frac{1}{2}\times 15\times 20 makes the area 150.

Area = 1/2 × 15 × 20 = 150
3STEP 3

Match the smallest altitude to the longest side

Any base ss gives h=300sh=\frac{300}{s}, and 300 never changes — so the longest side, the hypotenuse 25, carries the shortest altitude.

h = 300/s; larger s → smaller h; s_max=25
4STEP 4

Divide to get the height

Divide by the longest side: h=30025h=\frac{300}{25}, so the shortest altitude is 12 — choice (B).

h = 300/25 = 12 → (B)
Answer
12
Compute all three altitudes to be sure the smallest one was found: to side 15, h=300/15=20; to side 20, h=300/20=15; to side 25, h=300/25=12. The three heights are 20, 15, 12, and 12 is indeed the smallest, matching choice (B). It also makes sense that the altitude to the hypotenuse (12) is shorter than either leg (15 and 20), since a leg is itself the altitude to the other leg. Choice (E) 15 is the trap for anyone who stops at a leg instead of the hypotenuse.
💡Key takeaway

Since base times height is a fixed number (twice the area), the shortest height always drops to the longest side — so find the area, then divide by the biggest side.

  • Spot the right triangle
  • Find the area
  • Match the smallest altitude to the longest side
  • Divide to get the height

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