Competition · AMC preparation · step 4 of 4
AMC 8 · 2004 · #24
Grade 8 geometry-2d
Pick an answer.
AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The parallelogram is tilted, so measuring it directly is painful. Tool #16 (Count the Complement) flips the question: instead of measuring EFGH, measure the four corner right triangles that the rectangle gives up to form it, then subtract from the rectangle's area. That gives the parallelogram's area without any tilted measurements. Tool #7 (Identify Subproblems) splits the job in two — first find the area, then find the base FG — and Tool #1 (Draw a Diagram) lets us read each corner triangle's two leg lengths straight off the labeled sides. With the area and the base in hand, d comes from area = base × height.
Read the corner triangles
Each cut-off corner is a right triangle whose two legs are the labeled pieces — A: 4,3; B: 6,5; C: 3,4; D: 6,5.
Each corner sits at a 90° angle of the rectangle, so the labeled pieces on the two adjacent sides are exactly the legs — Grade 4 "identify right angles in figures".
4.G.A.1Draw A DiagramAdd the corner areas
The four corners pair up as two 3-4 and two 5-6 right triangles, so their combined area is 42.
Two pairs of congruent right triangles let you double the area of one from each pair — a Grade 6 right-triangle-area shortcut.
6.G.A.1Identify SubproblemsSubtract from the rectangle
The parallelogram is what remains after the corners are removed, so its area is 80 - 42 = 38.
"Whole minus the easy pieces" gives the tilted parallelogram's area without measuring any tilted side.
The inscribed parallelogram EFGH has area 38 — the rectangle's area 80 minus the four corner right triangles, which together cover 42.
▸ Why?
The rectangle is cut into exactly the parallelogram and the four corner triangles with no gaps and no overlaps, so those pieces add back to the rectangle, and the parallelogram is what is left once the four triangles are taken away.
▸ Why?
The rectangle measures 10 across and 8 up, and a 10-by-8 rectangle holds 10 rows of 8 unit squares, so its area is 80.
▸ Why?
The four corner pieces are right triangles sitting in the rectangle's square corners, so each has area half the product of its two labeled legs: 1/2(4)(3) + 1/2(6)(5) + 1/2(3)(4) + 1/2(6)(5) = 6 + 15 + 6 + 15 = 42.
▸ Why?
The two legs of such a right triangle frame a rectangle whose area is leg times leg unit squares.
▸ Why?
The triangle's long side is that rectangle's diagonal, and the diagonal turns the rectangle into two triangles that fall exactly onto each other under a half-turn, so each triangle is half the rectangle.
Find the base FG
Triangle CFG is a 3-4-5 right triangle, so the base FG equals its hypotenuse 5.
Grade 8 Pythagorean theorem on a 3-4-5 triangle gives an integer side — the cleanest possible base for the parallelogram.
8.G.B.7Identify SubproblemsDivide to get the height
Set the two area expressions equal: FG × d = 38, so 5d = 38 and d = 7.6.
One area, two expressions: equating them turns the unknown height d into a one-step division.
6.G.A.1Change Focus Count The ComplementThe tilted parallelogram is hard to measure directly, but the four corner triangles the rectangle gives up are easy: subtract their combined area 42 from the rectangle's 80 to get area 38, then divide by the base FG = 5 (a 3-4-5 triangle) to land on d = 7.6.
- Read the corner triangles
- Add the corner areas
- Subtract from the rectangle
- Find the base FG
- Divide to get the height
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