AMC 10 · 2004 · #20

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

equilateral-trianglepythagorean-theoremquadratic-equations convert-to-algebrasymmetry-argument ↑ Prerequisites: pythagorean-theorem

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

A square ABCD has two points E and F placed on its sides so that triangle BEF is equilateral, with E on side AD and F on side DC. Find the ratio of the area of triangle DEF to the area of triangle ABE.

Pick an answer.

(A)

$\frac{4}{3}$

(B)

$\frac{3}{2}$

(C)

$\sqrt{3}$

(D)

(E)

$1+\sqrt{3}$

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The one length that controls everything is how far E sits from corner A, so name it x. Every other length in the figure then becomes 1 or something built from x, and the equal sides of the equilateral triangle turn into equations through the Pythagorean theorem. The clever part at the end is that the area ratio can be simplified using the equation for x directly, so x never has to be solved for.

1STEP 1

Fix the square and name one length

Set the square's side to $1$ and let $AE = x$ ; the rest of that side is $DE = 1 - x$ .

AB = 1, AE = x, DE = 1 - x

2STEP 2

Match the two side lengths from B

Right angles at A and C give $1 + x^2 = BE^2 = BF^2 = 1 + CF^2$ , so $CF = x$ and $DF = DE = 1 - x$ .

BE² = 1 + x², BF² = 1 + CF² → CF = x → DF = DE = 1 - x

3STEP 3

Set the third side equal too

Triangle DEF is right-angled at D with equal legs, so $EF^2 = 2(1-x)^2$ , which must equal $BE^2 = 1 + x^2$ .

EF² = 2(1 - x)² = 1 + x²

4STEP 4

Simplify to a usable relation

Expanding gives $x^2 - 4x + 1 = 0$ , best kept as $x^2 + 1 = 4x$ — there is no need to solve for x.

2 - 4x + 2x² = 1 + x² → x² - 4x + 1 = 0 → x² + 1 = 4x

5STEP 5

Compare the two areas

Areas $\frac{x}{2}$ and $\frac{(1-x)^2}{2}$ give the ratio $\frac{(1-x)^2}{x} = \frac{(x^2+1) - 2x}{x} = \frac{2x}{x}$ , or 2 — choice (D).

[DEF]/[ABE] = (1/2(1-x)²)/(1/2 x) = ((1-x)²)/x = ((x²+1) - 2x)/x = (4x - 2x)/x = 2

Answer

The relation x squared minus 4x plus 1 equals 0 gives x = 2 minus root 3, about 0.27, which is a believable small distance up the side of a unit square. Plugging back, triangle ABE has area about 0.13 and triangle DEF has area about 0.27, whose quotient is 2, matching the tidy answer. Since DEF spans nearly the full top-left corner while ABE hugs the bottom-left corner, DEF being about twice as large is sensible.

💡Key takeaway

Name the one unknown length, turn the equal sides into an equation, then feed that equation into the area ratio so the unknown cancels.

Fix the square and name one length
Match the two side lengths from B
Set the third side equal too
Simplify to a usable relation
Compare the two areas

Fix the square and name one length

Match the two side lengths from B

Set the third side equal too

Simplify to a usable relation

Compare the two areas

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