AMC 10 · 2016 · #6
Grade 8 geometry-2dPick an answer.
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
The level side makes the ends symmetric.
Squaring erases the sign, so a parabola is a perfect mirror across the y-axis.
Squaring erases the sign, so the curve is a perfect mirror across the vertical axis.
▸ Why?
A number and its opposite have the same square, so both give the same height.
▸ Why?
Reflecting across that axis carries the curve onto itself without stretching anything.
Read off the base and height
The base and height are then read off.
Horizontal length is the gap in x; the height is the gap in y up to the base.
6.G.A.3Introduce A VariableWrite the area equation
The area becomes a plain cube.
The 1/2 and the 2 cancel, leaving the tidy fact that the area equals a³.
6.G.A.1Introduce A VariableSolve and find BC
Solving gives 8, choice (C).
A cube root undoes a cube, so a³ = 64 points straight to a = 4.
8.EE.A.2Introduce A VariableA 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