AMC 10 · 2006 · #8
Grade 8 geometry-2d
Pick an answer.
AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The area formula for a semicircle needs the radius, and the radius is hidden inside the picture, so Tool #1 (Draw a Diagram) marks the center at the middle of the diameter and draws the straight line from that center out to a top corner of the square. Tool #4 (Introduce a Variable) names the square's half-side so the corner's position can be written down, and Tool #7 (Identify Subproblems) splits the job into three clean pieces: turn the area into a side length, use a right triangle to get the radius, then plug the radius into the circle-area formula.
Mark the center and a half-side
Put the center O at the diameter's midpoint and call half the base a, so the side is 2a and a top corner sits 2a above.
Placing the center at the middle turns the symmetric picture into two equal halves, so one small length a describes the whole square.
6.G.A.3Draw A DiagramTurn the area into a fact about a
The side is 2a, so the area is (2a)² = 4a² = 40, giving a² = 10 — and a² is all we ever need.
Keeping the answer as a² instead of a avoids messy roots, because the radius will also come out squared.
6.EE.B.7Introduce A VariableGet the radius with a right triangle
The radius to a top corner is the hypotenuse over legs a and 2a, so r² = a² + 4a² = 5a² = 50.
The radius, the half-base, and the side form a right triangle, so the Pythagorean Theorem links them directly.
The radius, the half-base, and the side form a right triangle, so one equation links them.
▸ Why?
With that right angle the square on the longest side equals the two shorter squares added.
▸ Why?
Every point of the circle sits the same distance from the centre, so that distance is the radius everywhere.
Apply the circle-area formula
A semicircle is half a circle, so its area is half of π r², that is half of π(50) = 25π — choice (B).
Because r² is already known, the area formula needs no square root at all.
7.G.B.4Identify SubproblemsTo find a circle's size, hunt for a right triangle that reaches from the center to a point on the edge — the Pythagorean Theorem hands you the radius squared, which is all the area formula needs.
- Mark the center and a half-side
- Turn the area into a fact about a
- Get the radius with a right triangle
- Apply the circle-area formula