AMC 10 · 2016 · #9
Grade 8 geometry-2dPick an answer.
AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The parabola y = x² is a mirror image across the y-axis: if a point (x, x²) is on it, so is (-x, x²). Since BC is horizontal, B and C sit at the same height, so they must be that mirror pair. Tool #1 (Draw a Diagram) makes the symmetry visible, and Tool #4 (Introduce a Variable) lets us call the half-width a so the base and height both become simple expressions in a. Then the area condition is one equation in one unknown.
Place B and C using symmetry
BC is horizontal, so B and C share a height; on y = x² equal heights force opposite x-values, giving B = (-a, a²), C = (a, a²).
Squaring erases the sign, so a parabola is a perfect mirror across the y-axis.
6.G.A.3Draw A DiagramRead off the base and height
The base spans x = -a to a, so BC = 2a; A sits at height 0 and BC at height a², so the triangle's height is a².
Horizontal length is the gap in x; the height is the gap in y up to the base.
6.G.A.3Use Matrix LogicWrite the area equation
Area is half the base times the height: ½·(2a)·a² simplifies to a³ = 64, one equation in one unknown.
The 1/2 and the 2 cancel, leaving the tidy fact that the area equals a³.
6.G.A.1Use Matrix LogicSolve and find BC
Cube-root a³ = 64 to get a = 4, so BC = 2a = 8; the answer is (C).
A cube root undoes a cube, so a³ = 64 points straight to a = 4.
8.EE.A.2Use Matrix LogicA parabola is a mirror, so two points at the same height sit at ± a; then the area is just a³, and a cube root finishes it.
- Place B and C using symmetry
- Read off the base and height
- Write the area equation
- Solve and find BC