AMC 10 · 2009 · #4
Grade 6 geometry-2d
Pick an answer.
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
The difference of the parallel sides gives each triangle's leg.
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
The two triangles combine into one square.
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?
Turning one triangle onto the other moves it without stretching, so the joined edges match exactly.
▸ Why?
Each triangle is half the square on those legs, so the two together fill it exactly.
Find the whole yard's area
The same leg is the rectangle's height.
A rectangle's area is just its length times its width.
4.MD.A.3Identify SubproblemsDivide and simplify
Dividing and reducing gives 1/5, 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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