AMC 10 · 2007 · #14

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
integer-pythagorean-triplespythagorean-theoremsimilar-trianglesarea-triangles identify-subproblems ↑ Prerequisites: pythagorean-theorem
📏 Medium solution 💡 3 insights
Problem
A triangle whose side lengths are in the ratio 3:4:5 is drawn inside a circle of radius 3 so that all three corners touch the circle. Find the area of this triangle.

Pick an answer.

(A)
$\ 8.64$
(B)
$\ 12$
(C)
$\ 5\pi$
(D)
$\ 17.28$
(E)
$\ 18$

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The whole problem turns on one picture, so Tool #1 (Draw a Diagram) is primary: sketching the triangle inside the circle exposes the key fact that its longest side is a diameter. Getting there needs Tool #5 (Look for a Pattern) to recognize 3:4:5 as the famous Pythagorean triple, which makes the triangle a right triangle. Then Tool #4 (Introduce a Variable) sets a single scale factor k so the ratio sides 3,4,5 become real lengths, and Tool #7 (Identify Subproblems) splits the job into clean stages — prove it is right, find the diameter, scale the legs, then take the area.

1STEP 1

Recognize the 3-4-5 right triangle

3²+4²=25=5², so by the converse of the Pythagorean theorem the triangle is right-angled, with its longest side facing the right angle.

3²+4² = 9+16 = 25 = 5² → right triangle
2STEP 2

The hypotenuse is a diameter

A right angle sitting on a circle always faces a diameter, so the hypotenuse runs through the center: it is 2r = 6.

hypotenuse = diameter = 2r = 2 · 3 = 6
3STEP 3

Scale the ratio up to real lengths

Write the sides as 3k, 4k, 5k; since 5k = 6, the scale factor is k = 1.2 and the legs become 3.6 and 4.8.

5k = 6 → k = 6/5 = 1.2; 3k = 3.6, 4k = 4.8
4STEP 4

Take half of leg times leg

Perpendicular legs act as base and height, so the area is half their product: half of 3.6 × 4.8 = 8.64, choice (A).

Area = 1/2(3.6)(4.8) = 1/2(17.28) = 8.64 → (A)
Answer
8.64
The circle's area is π r² = 9π ≈ 28.3, and the triangle sits inside it, so its area must be well under 28, and 8.64 is comfortably less. A sharper check: the plain 3-4-5 triangle has area 1/2(3)(4)=6, and scaling every length by k=1.2 multiplies area by k²=1.44, giving 6 × 1.44 = 8.64 — the same answer two ways. The trap answer (D) 17.28 is exactly 3.6 × 4.8, i.e. forgetting the factor of 1/2; (C) 5π≈ 15.7 wrongly expects a π (areas of triangles carry no π); (B) 12 and (E) 18 don't come from any correct step.
💡Key takeaway

A 3-4-5 triangle is a right triangle, so its longest side stretches across the circle as the diameter; scale everything to match, then take half of leg times leg.

  • Recognize the 3-4-5 right triangle
  • The hypotenuse is a diameter
  • Scale the ratio up to real lengths
  • Take half of leg times leg

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