AMC 10 · 2016 · #21
Grade 8 geometry-2dPick an answer.
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.
Set up a symmetric coordinate frame
Symmetry lets a frame be chosen cleanly.
Equal neighboring sides mean the picture looks the same in a mirror, so a symmetry axis through the center is the cheapest place to anchor coordinates.
6.G.A.3Draw A DiagramPlace the two top vertices
The circle places the two top corners.
A chord, half its length, and the perpendicular from the center always form a right triangle, so the Pythagorean theorem hands you the height.
A chord, half its length, and the perpendicular from the centre always form a right triangle.
▸ Why?
The centre is equally far from both ends of the chord, so its perpendicular lands exactly in the middle.
▸ Why?
With that right angle, the radius, the half-chord and the distance are tied by one equation.
Write A's two conditions and reduce them
The last corner has two conditions.
Subtracting the circle equation from the distance equation cancels the squared terms, leaving one tidy line that A must sit on.
8.G.B.8Identify SubproblemsSolve for A's coordinates
Combining them leaves one quadratic.
Plugging the line into the circle leaves one equation in one unknown, and the radical arithmetic stays clean because every square root is a multiple of √(7).
8.EE.A.2Introduce A VariableRead off the fourth side
Reading the width gives 500, choice (E).
Once both endpoints are points on the grid, the side length is just the distance between them, which here is a plain horizontal gap.
8.G.B.8Introduce A VariableThree 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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