AMC 10 · 2006 · #21
Grade 11 geometry-2dPick an answer.
Four lengths matter here — the rectangle's two sides and the ellipse's two semi-axes — so Tool #4 (Introduce a Variable) names them l, w, a, b and lets the two area facts become two equations. Tool #1 (Draw a Diagram) is what reveals the bridge: the corner A sits at the end of both sides AB and AD, and those are exactly the two distances from A to the foci, while BD is both the rectangle's diagonal and the focus-to-focus distance. Tool #15 (Organize Information in More Ways) is the decisive move — the perimeter needs only l+w, never l and w separately, so the whole problem should be rewritten in terms of the sum l+w and the product lw. Tool #13 (Convert to Algebra) then grinds those relations together. Finally Tool #6 (Guess and Check) is used honestly at the end: the equations alone only say what the perimeter must be if such a rectangle exists, so an actual pair of side lengths has to be produced.
Name the four lengths
The two given areas become two products.
Two different shapes each hand over one equation, and both happen to say that a product of two lengths equals 2006.
9.A-CED.A.2Introduce A VariableCorner A gives half the perimeter
The defining property makes the perimeter a multiple of one axis.
The two sides meeting at a corner are exactly that corner's two distances to the far corners, so the focal-sum rule hands over the semi-perimeter for free.
11.G-GPE.A.3Draw A DiagramThe diagonal is the focal distance
The diagonal is exactly the focal distance.
The foci sit at opposite corners, so the gap between them is just the rectangle's diagonal.
8.G.B.7Draw A DiagramThe crux: b² is half the rectangle's area
The ellipse relation makes the other axis squared half the rectangle's area.
Squaring a sum and subtracting the sum of squares leaves only the cross term, so the shape information cancels and the area is all that is left.
Squaring a sum and subtracting the sum of squares leaves only the cross term, so the area falls out on its own.
▸ Why?
Expanding a sum multiplied by itself gives each square once and the cross term twice.
▸ Why?
The diagonal squared is the two sides squared added together, which is exactly the sum being subtracted.
The ellipse's area now pins down a
The given ellipse area then pins the first axis.
Once the short semi-axis is known, the fixed area forces the long one, and the long one turns out to be exactly twice the short one.
8.EE.A.2Convert To AlgebraCheck the rectangle exists, then read off the perimeter
A real rectangle exists, so the perimeter is 8√(1003), choice (C).
Equations can describe an impossible shape, so producing the actual side lengths turns a forced value into a real answer.
9.A-REI.B.4Guess And CheckThe two sides meeting at a corner are that corner's distances to the two foci, so the semi-perimeter is the ellipse's 2a; and because (l+w)²-(l²+w²)=2lw, the short semi-axis knows only the rectangle's area, which pins the perimeter at 8√(1003).
- Name the four lengths
- Corner A gives half the perimeter
- The diagonal is the focal distance
- The crux: b² is half the rectangle's area
- The ellipse's area now pins down a
- Check the rectangle exists, then read off the perimeter