AMC 10 · 2022 · #13

Grade 8 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

pythagorean-theoremsimilar-trianglescoordinate-geometryarea-trianglesarea-rectangles identify-subproblemsarea-differencecomplementary-counting ↑ Prerequisites: pythagorean-theoremsimilar-triangles

📏 Long solution 💡 3 insights 📊 Diagram

Problem

A four by eight rectangle contains a square of side 5. Three corners of the square sit on three different sides of the rectangle. Find the area of the region lying inside both shapes.

Pick an answer.

(A)

$15\dfrac{1}{8}$

(B)

$15\dfrac{3}{8}$

(C)

$15\dfrac{1}{2}$

(D)

$15\dfrac{5}{8}$

(E)

$15\dfrac{7}{8}$

How to solve

Strategy Draw a Diagram

Tool #1 (Diagram): drop the picture on coordinates so the rectangle is [0,8] × [0,4]. Tool #7 (Subproblems): the two right triangles formed by the square's slanted sides and the bottom/right edges of the rectangle are both 3-4-5 — that fixes every vertex coordinate. Tool #16 (Complement): instead of computing the overlap directly (a pentagon), compute the whole square (25) and subtract the small triangle that pokes above the rectangle's top edge.

1STEP 1

Use the right angle and equal sides

Adjacent sides are perpendicular and equal.

AB = BC = 5, ∠ ABC = 90°

2STEP 2

Find one piece of a side

Pythagoras gives the missing piece.

A'B = √(25 - 16) = 3

3STEP 3

Fix the three corners

The three corners' coordinates are fixed.

A = (1, 4), B = (4, 0), C = (8, 3)

4STEP 4

Find the fourth corner

The parallelogram rule gives the fourth.

D = (1+4, 4+3) = (5, 7)

5STEP 5

Locate what pokes outside

Only one corner pokes outside.

G = (29/4, 4)

6STEP 6

Area of the outside piece

The outside piece is a triangle.

outside area = 1/2 · 25/4 · 3 = 75/8

7STEP 7

Subtract to finish

Subtracting gives fifteen and five eighths.

25 - 75/8 = 125/8 = 15 5/8

Answer

15 5/8

The overlap must be less than the square (25) and less than the rectangle (32). 15 5/8 sits comfortably below both. The spilled triangle has area 75/8 ≈ 9.4, which feels right for a triangle of base ≈ 6.25 and height 3. Also, the answer choices are tightly bunched between 15 1/8 and 15 7/8 — only careful arithmetic on the eighth (75/8) picks the right one.

💡Key takeaway

Two hidden 3-4-5 right triangles fix every corner of the square. The square's top corner (5, 7) pokes above the rectangle, so chop off that little triangle — area 75/8 — and the leftover is 25 - 75/8 = 15 5/8, choice (D).