AMC 10 · 2003 · #11
Grade 8 geometry-2dPick an answer.
Nothing in the problem has a number attached, so tool #4 (Introduce a Variable) supplies the missing handle: name the shared perimeter P and every length in the picture becomes an expression in P. Tool #7 (Identify Subproblems) then splits the work into two independent radius hunts, one for the square and one for the triangle, with the ratio assembled only at the very end. Tool #1 (Draw a Diagram) is what makes each radius findable: drawing the center and dropping the right segments turns both radius questions into right triangles, and right triangles are something the Pythagorean theorem can finish. Because only a ratio is wanted, P and π are expected to cancel, which is also a built-in check that the setup was consistent.
Name the perimeter and get the sides
With shared perimeter P, the sides are P/4 and P/3 — different, since the shapes have different side counts.
Equal perimeters split unequally: the shape with fewer sides has to make each one longer.
6.RP.A.3Introduce A VariableFind the square's circle radius
A square's diagonal is a full diameter, giving radius squared P²/32.
A chord through the center is a diameter, so the square's diagonal hands you the circle's size for free.
The square's diagonal passes through the centre, so it is a diameter and hands over the circle's size at once.
▸ Why?
Every point of a circle sits one radius from the centre, so a chord through the centre is two radii long.
▸ Why?
The diagonal is the hypotenuse of the right triangle made by two sides, so its square is the two squares added.
Find the triangle's altitude
For the triangle, dropping the altitude gives height s√(3)/2.
Cutting a symmetric shape down its mirror line always produces a right triangle you can measure.
8.G.B.7Draw A DiagramFind the triangle's circle radius
Placing the centre on that altitude gives radius squared P²/27.
Demanding that the center be equally far from two different vertices pins it down, and the squared unknown cancels itself out.
8.EE.C.7Identify SubproblemsDivide the two areas
Dividing cancels pi and the perimeter, leaving 27/32, choice (C).
Once both circles are described by the same letters, the letters cancel and only the pure number survives.
7.G.B.4Identify SubproblemsWhen two shapes share a perimeter, give that perimeter a letter, use right triangles inside each shape to find its circle's radius, and let the shared letters cancel when you divide.
- Name the perimeter and get the sides
- Find the square's circle radius
- Find the triangle's altitude
- Find the triangle's circle radius
- Divide the two areas