AMC 10 · 2003 · #6

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
pythagorean-theoremratio-proportioninteger-pythagorean-triples convert-to-algebra ↑ Prerequisites: pythagorean-theoremratio-proportion
📏 Medium solution 💡 2 insights
Problem
A television screen is a rectangle whose horizontal length and height are in the ratio 4:3. The screen is called "27-inch" because its diagonal measures 27 inches. Find which answer choice is closest to the horizontal length, in inches.

Pick an answer.

(A)
20
(B)
20.5
(C)
21
(D)
21.5
(E)
22

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The words "horizontal," "height," and "diagonal" describe a shape, so Tool #1 (Draw a Diagram) comes first: drawing the diagonal splits the rectangle into a right triangle whose legs are the length and the height and whose hypotenuse is the 27-inch diagonal. Tool #4 (Introduce a Variable) turns the ratio 4:3 into legs 4x and 3x, so one unknown x describes both sides at once. Tool #5 (Look for a Pattern) spots that legs in the ratio 3:4 make a 3-4-5 right triangle, so the hypotenuse is 5x — no square roots needed, just solve 5x=27.

1STEP 1

Draw the diagonal, make a right triangle

Draw the diagonal: it cuts the rectangle into right triangles with legs the length and the height, and the 27-inch diagonal as hypotenuse.

legs = length, height; hypotenuse = 27
2STEP 2

Name the sides with the ratio

Since length : height = 4:3, write the length as 4x and the height as 3x, so one positive unknown x describes both sides.

length = 4x, height = 3x
3STEP 3

Use the 3-4-5 triangle

Legs 3x and 4x are a 3-4-5 triangle scaled by x, since (3x)²+(4x)²=25x²=(5x)², so the hypotenuse gives 5x=27.

(3x)²+(4x)²=25x²=(5x)² → 5x=27
4STEP 4

Solve for the length

Divide 5x=27 by 5 to get x=5.4, so the horizontal length is 4x=21.6 inches — the nearest choice is 21.5, answer (D).

x=27/5=5.4; 4x=4×5.4=21.6 ≈ 21.5 → (D)
Answer
21.5
The horizontal length must be less than the diagonal of 27 inches but more than the height, so a value in the low 20s is sensible — 21.6 fits. A quick sanity check: the height is 3x=16.2 inches, and 21.6²+16.2²=466.56+262.44=729=27², so the sides really do give a 27-inch diagonal. The trap is treating 27 as a side instead of the diagonal; then 4 of 7 parts of 27 would be about 15.4, which is not even a choice, confirming the diagonal reading is right.
💡Key takeaway

The diagonal of a 4:3 screen makes a 3-4-5 triangle, so the diagonal is 5 parts — split it into 5 and take 4 of them for the width.

  • Draw the diagonal, make a right triangle
  • Name the sides with the ratio
  • Use the 3-4-5 triangle
  • Solve for the length

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