AMC 10 · 2017 · #8

Grade 9 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
pythagorean-theoremquadratic-equationsratio-proportiongolden-ratio convert-to-algebraidentify-subproblems ↑ Prerequisites: pythagorean-theoremquadratic-equations
📏 Medium solution 💡 2 insights
Problem
A rectangle has a short side, a long side, and a diagonal. The short side divided by the long side gives the same number as the long side divided by the diagonal. Find the square of the first of those two equal ratios.

Pick an answer.

(A)
$\frac{\sqrt{3}-1}{2}$
(B)
$\frac{1}{2}$
(C)
$\frac{\sqrt{5}-1}{2}$
(D)
$\frac{\sqrt{2}}{2}$
(E)
$\frac{\sqrt{6}-1}{2}$
How to solve
Strategy Introduce a Variable

No lengths are given, and none can be found: doubling every length leaves the condition unchanged. So chasing a and b separately is chasing something that does not exist. The fix is to name the quantity the problem actually asks for, k = (a/b)², and force the whole equation to be about k alone. That works because after using the Pythagorean theorem every term in the equation is a product of four length factors, so dividing through by b⁴ leaves pure ratios and no leftover units. What comes out is a single quadratic in k, and a quick sign check picks the one root a squared ratio is allowed to be.

1STEP 1

Name the sides, find the diagonal

The diagonal follows directly from the two sides.

d = √(a² + b²)
2STEP 2

Write the ratio sentence as an equation

The equal-ratio sentence becomes one equation.

a/b = b/d ⟹ b² = ad = a√(a² + b²)
3STEP 3

Make the answer the unknown

Making the target the unknown turns it quadratic.

b⁴ = a²(a² + b²) → 1 = k(k+1), k = a²/b²
4STEP 4

Solve and keep the possible root

Dropping the negative root leaves the golden ratio's reciprocal.

k = (-1 ± √(5))/2, k ＞ 0 ⟹ k = (√(5)-1)/2
Answer
(√(5)-1)/2
Test the five choices against the equation k² + k = 1 directly, no algebra needed. (A) (√(3)-1)/2 ≈ 0.366 gives 0.500. (B) 1/2 gives 0.750. (C) (√(5)-1)/2 ≈ 0.618 gives 0.382 + 0.618 = 1.000. (D) √(2)/2 ≈ 0.707 gives 1.207. (E) (√(6)-1)/2 ≈ 0.725 gives 1.250. Only (C) hits 1. Rebuilding the rectangle confirms it: with k ≈ 0.618 the ratio a/b = √(k) ≈ 0.7862, so taking a = 1 gives b ≈ 1.2720 and d = √(1 + 1.6180) = √(2.6180) ≈ 1.6180. Then a/b ≈ 0.7862 and b/d = 1.2720/1.6180 ≈ 0.7862, the two ratios agree, and 1 ＜ 1.2720 ＜ 1.6180 confirms the short side really is shorter than the long side and both are shorter than the diagonal. The answer also had to be less than 1, since a short side over a long side is less than 1, and 0.618 is.
💡Key takeaway

When a problem only fixes a shape and not a size, name the ratio it asks for as one letter and divide the whole equation by a matching power of one side; the lengths cancel and a single equation is left.

  • Name the sides, find the diagonal
  • Write the ratio sentence as an equation
  • Make the answer the unknown
  • Solve and keep the possible root

More like this

Same archetype — closest grade level first.