AMC 10 · 2009 · #4
Grade 6 geometry-2d
Pick an answer.
AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The figure is a compound shape, so Tool #7 (Identify Subproblems) splits the work into three clean pieces: first find the triangle leg length from the two parallel sides, then find the total flower-bed area, then find the whole-yard area — and divide. Tool #17 (Visualize Spatial Relationships) supplies the key shortcut: the two congruent right triangles slide together into a single square, so their combined area is easy. Tool #1 (Draw a Diagram) reads the leg length straight off the picture, and Tool #3 (Eliminate Possibilities) checks the final fraction against the choices.
Find each triangle's leg
Bottom is the full 25 and the top has 15 between the beds, so the two legs share 25-15=10: each leg is 5 — also the yard's height.
An isosceles right triangle has two equal legs, so once you know one leg you know the other — and the height of the yard.
4.G.A.2Draw A DiagramCombine the two triangles into a square
Slide the two congruent triangles together and they form a 5×5 square, so the beds cover 25 square meters in all.
Two congruent right triangles glued along their equal legs always make a square, turning a hard area into an easy one.
Two matching right triangles glued along their equal legs always make a square.
▸ Why?
Two equal legs make the two other angles equal, so each is half a right angle and the corners fit.
▸ Why?
Flipping one triangle onto the other moves it without stretching it, so no length changes.
Find the whole yard's area
The yard is a rectangle 25 long and 5 wide (the leg from step 1), so its area is 125 square meters.
A rectangle's area is just its length times its width.
4.MD.A.3Identify SubproblemsDivide and simplify
The beds take 25/125 of the yard; dividing top and bottom by 25 gives 1/5, which is choice (C).
Dividing numerator and denominator by the same number keeps the fraction's value while shrinking it to lowest terms.
4.NF.A.1Eliminate PossibilitiesTwo matching right triangles snap together into a square, so find that square's area, compare it to the whole rectangle, and reduce the fraction.
- Find each triangle's leg
- Combine the two triangles into a square
- Find the whole yard's area
- Divide and simplify
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