AMC 10 · 2016 · #24

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
arc-measurelaw-of-cosinesisosceles-triangle convert-to-algebra ↑ Prerequisites: law-of-cosines
📏 Long solution 💡 4 insights
Problem
A four-sided figure has all four corners sitting on one circle whose radius is 200√(2). Three of its sides each have length 200. Find the length of the remaining (fourth) side.

Pick an answer.

(A)
200
(B)
$200\sqrt{2}$
(C)
$200\sqrt{3}$
(D)
$300\sqrt{2}$
(E)
500

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Three equal consecutive chords make the figure mirror-symmetric: tool #17 (Visualize Spatial Relationships) spots that reflecting across the line through the center and the middle of the equal sides swaps the two end vertices, so the whole picture has a clean axis. That makes tool #4 (Introduce a Variable) the natural engine — put the center at the origin and the mirror line as the y-axis, and every vertex becomes a coordinate I can pin down with the circle equation x²+y²=R². Tool #1 (Draw a Diagram) fixes which vertices are neighbors so I know which chords must equal 200. Tool #7 (Identify Subproblems) splits the job into easy stages: locate the two top vertices from the equal side, then locate one bottom vertex, then read off the fourth side from symmetry.

1STEP 1

Set up a symmetric coordinate frame

Order it ABCD with AB=BC=CD=200; the equal sides force mirror symmetry, so center O at the origin, y-axis as mirror, giving x²+y²=80000.

AB=BC=CD=200, x²+y²=(200√(2))²=80000
2STEP 2

Place the two top vertices

BC=200 is the top equal side, horizontal and centered, so B=(-100,h), C=(100,h); the chord right triangle gives h=100√(7).

h=√(R²-100²)=√(80000-10000)=√(70000)=100√(7)
3STEP 3

Write A's two conditions and reduce them

A=(x,y) lies on x²+y²=80000 and is 200 from B; subtracting the circle equation cancels the squares, leaving the line x-√(7) y=-600.

(x+100)²+(y-100√(7))²=40000 → x-√(7) y=-600
4STEP 4

Solve for A's coordinates

Put the line into the circle: y²-150√(7) y+35000=0 gives y=100√(7) (that is C) or y=50√(7), so A=(-250, 50√(7)).

y²-150√(7) y+35000=0 → y=50√(7), x=-250
5STEP 5

Read off the fourth side

By symmetry D=(250, 50√(7)) mirrors A, so the horizontal side DA=250-(-250)=500, far longer than the three 200 sides — answer (E).

DA=250-(-250)=500 → (E)
Answer
500
Check that A truly lies on the circle: (-250)²+(50√(7))²=62500+17500=80000=R², and that AB=200: (-250+100)²+(50√(7)-100√(7))²=(-150)²+(-50√(7))²=22500+17500=40000=200². Both hold. The size also makes sense: three short chords of 200 each cover only small arcs near the top, leaving a big bottom arc whose chord must be the longest side, and 500 is the largest choice, larger than 200√(2)≈283 and 300√(2)≈424.
💡Key takeaway

Three equal sides make the shape mirror-symmetric, so drop it on a grid with the center at the origin, find each corner with the circle equation, and the long fourth side is just the distance across the bottom — 500.

  • Set up a symmetric coordinate frame
  • Place the two top vertices
  • Write A's two conditions and reduce them
  • Solve for A's coordinates
  • Read off the fourth side

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