AMC 8 · 2005 · #9

Grade 8 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

isosceles-triangleangle-sum-triangle identify-subproblems ↑ Prerequisites: angle-sum-triangleisosceles-triangle

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

In quadrilateral ABCD, the two sides meeting at B have length AB = BC = 10, and the two sides meeting at D have length CD = DA = 17. The angle at D is ∠ ADC = 60°. Find the length of diagonal AC.

Pick an answer.

(A)

13.5

(B)

(C)

15.5

(D)

(E)

18.5

AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Tool #1 (Draw a Diagram) shows that the diagonal AC slices the quadrilateral into two triangles, △ ABC and △ ADC. Only one of them carries useful given information: △ ADC has two known sides (DA = CD = 17) and the angle between them (60°). The sides AB and BC at vertex B are a distraction. Tool #7 (Identify Subproblems) reduces the original quadrilateral question to a single subproblem about △ ADC: find its third side AC. The isosceles structure plus the 60° apex angle forces all three angles to be 60°, making the triangle equilateral, so AC = 17 with no calculation beyond the angle sum.

1STEP 1

Draw the diagonal AC; it splits ABCD into △ ABC and △ ADC, and every given measurement sits inside △ ADC.

△ ADC: DA = 17, CD = 17, ∠ ADC = 60°

2STEP 2

In △ ADC two sides are equal (DA = CD = 17), so it is isosceles and its two base angles ∠ DAC = ∠ DCA.

DA = CD → ∠ DCA = ∠ DAC

3STEP 3

By the 180° angle sum, x + x + 60° = 180°, so each base angle x = 60°.

x + x + 60 = 180 → 2x = 120 → x = 60

4STEP 4

All three angles are 60°, so △ ADC is equilateral — every side equal, giving AC = DA = CD = 17.

∠ ADC = ∠ DAC = ∠ DCA = 60° → AC = 17 → (D)

Answer

The answer AC = 17 is one of the listed choices and exactly matches DA = CD = 17, which fits the equilateral conclusion. It is also between the longest pair of sides (17) and the shortest pair (10): the diagonal of a convex quadrilateral has to be shorter than the sum of two adjacent sides (17 + 17 = 34 and 10 + 10 = 20) and longer than their difference (17 - 10 = 7), which 17 comfortably satisfies. The fact that the 10s at vertex B were never used is a hint that the problem really did want only the △ ADC subproblem.

💡Key takeaway

The diagonal AC lives inside two triangles, but only one of them carries the given 17, 17, and 60°. That isosceles triangle with a 60° apex angle has to be equilateral, so the third side just equals 17.

More like this