AMC 10 · 2010 · #7

Grade 8 geometry-2d
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
area-trianglespythagorean-theoremarea-rectanglesperimeter identify-subproblems ↑ Prerequisites: area-trianglespythagorean-theorem
📏 Medium solution 💡 2 insights
Problem
A triangle has sides 10, 10, and 12. A rectangle that is 4 wide covers the same area as this triangle. Find the perimeter of that rectangle.

Pick an answer.

(A)
16
(B)
24
(C)
28
(D)
32
(E)
36

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The perimeter cannot be reached in one move. Break it into a chain: find the triangle's height, then its area, then use that area with the known width to get the rectangle's length, and finally the perimeter. Each subproblem is small and standard.

1STEP 1

Split the triangle in half

The triangle is isosceles, so the height to the base 12 splits it into two right triangles with leg 6 and hypotenuse 10.

12 ÷ 2 = 6
2STEP 2

Find the height with Pythagoras

Pythagoras on that right triangle gives h squared equal to 100 minus 36, or 64, so the height is 8.

h² + 6² = 10² → h² = 100 - 36 = 64 → h = 8
3STEP 3

Compute the triangle's area

With base 12 and height 8, the triangle's area is half of 12 times 8, which is 48.

A_△ = 1/2 · 12 · 8 = 48
4STEP 4

Find the rectangle's length

The rectangle has the same area 48 and width 4, so its length is 48 divided by 4, which is 12.

4 · L = 48 → L = 48/4 = 12
5STEP 5

Add up the perimeter

The rectangle is 4 by 12, so the perimeter is 2 times the sum 4 plus 12, which is 32 — choice (D).

P = 2(4 + 12) = 2 · 16 = 32
Answer
32
The height 8 fits, since 6-8-10 is a well-known right triangle. Area 48 and width 4 give length 12, which is larger than the width, as expected for a long thin rectangle. Perimeter 2(4+12)=32 matches choice (D). A quick sanity check: the rectangle's length 12 equals the triangle's base 12, which makes sense because the rectangle's height 4 is exactly half the triangle's height 8.
💡Key takeaway

Chase the area from shape to shape: find the triangle's area, hand it to the rectangle, and let width divide out the missing side.

  • Split the triangle in half
  • Find the height with Pythagoras
  • Compute the triangle's area
  • Find the rectangle's length
  • Add up the perimeter

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