AMC 10 · 2003 · #5
Grade 8 rate-ratioPick an answer.
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.
Draw the diagonal, make a right triangle
The diagonal cuts the rectangle into right triangles with the diagonal as hypotenuse.
The diagonal of a rectangle is always the hypotenuse of a right triangle whose legs are the two sides.
4.G.A.2Draw A DiagramName the sides with the ratio
The ratio writes both sides with one unknown: 4x and 3x.
A ratio 4:3 means both sides grow together, so one multiplier x fixes both.
6.RP.A.3Introduce A VariableUse the 3-4-5 triangle
Those legs form a scaled 3-4-5 triangle, so the hypotenuse is 5x = 27.
Legs in the ratio 3:4 always give a hypotenuse 5 parts long, the classic 3-4-5 triangle.
Legs in the ratio three to four always give a hypotenuse five parts long.
▸ Why?
In a right triangle the square on the hypotenuse is the two squares on the legs added together.
▸ Why?
Scaling all three sides by one multiplier keeps the shape, so any 3-4-5 triangle is a resized copy of the smallest one.
Solve for the length
So x is 5.4 and the width is 21.6, closest to 21.5, choice (D).
Once x is known, the length is just four of those equal parts.
6.RP.A.3Introduce A VariableThe 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