AMC 10 · 2005 · #7
Grade 8 geometry-2dPick an answer.
Absolute value hides a shape behind a formula, so the fastest route is to get the shape on paper. The two sign symmetries mean one quadrant carries all the information: handle the easy first-quadrant piece, then let the symmetry rebuild the rest and the area follows by multiplication.
Pull the constants out
The constants come out, so only distance from zero matters.
Absolute value measures distance from zero, and a positive multiplier scales that distance without touching the sign.
6.NS.C.7Organize Information In More WaysSolve one quadrant first
In the positive quadrant it is an ordinary line segment.
Fixing the signs removes the absolute value bars, and what is left is a plain linear equation.
8.F.A.3Solve An Easier Related ProblemReflect to get the full shape
Reflecting gives a closed shape with diagonals on the axes.
The equation is blind to the signs of x and y, so whatever happens in one quadrant is mirrored into the other three.
The equation is blind to the signs, so whatever happens in one quadrant is mirrored into the other three.
▸ Why?
Reflecting across an axis moves the figure without stretching it, so each mirrored piece is the same size.
▸ Why?
A triangle cut off by the two axes has the intercepts as its legs, so its area is half their product.
Area of the corner triangle
The corner triangle has area 6.
A triangle cut off by the two axes has the axis intercepts as its legs, so its area is half their product.
6.G.A.1Identify SubproblemsMultiply by four
Four congruent copies give 24, choice (D).
Congruent pieces have equal area, so counting one piece and multiplying finishes the job.
6.G.A.1Identify SubproblemsAbsolute value bars fold a picture across both axes, so work out one quadrant and multiply by four.
- Pull the constants out
- Solve one quadrant first
- Reflect to get the full shape
- Area of the corner triangle
- Multiply by four