Problem 19 · 2015 AMC 8
Medium
Geometry & Measurement
shoelacecoordinate-geometry
A triangle with vertices as A = (1, 3), B = (5, 1), and C = (4, 4) is plotted on a 6 × 5 grid. What fraction of the grid is covered by the triangle?
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Answer: A — 1/6.
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Hint 1 of 2
The 'fraction of the grid' is just (triangle area) ÷ (grid area), and the grid is an easy 6 × 5 = 30. So the whole problem reduces to finding one triangle's area from its coordinates.
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Hint 2 of 2
A tilted triangle on a grid has no obvious base or height — that's exactly when the shoelace formula shines: it gets the area straight from the three coordinate pairs, no measuring.
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Approach: shoelace formula for area from coordinates
- Area = ½ |xA(yB − yC) + xB(yC − yA) + xC(yA − yB)| = ½ |1(1 − 4) + 5(4 − 3) + 4(3 − 1)| = ½ |−3 + 5 + 8| = 5.
- Fraction = 5 / 30 = 1/6.
- Why this transfers: shoelace turns any polygon's vertices into its area mechanically — the antidote to slanted shapes with no clean base.
Another way — bounding box minus corner triangles:
- Box the triangle in the rectangle from (1,1) to (5,4): area 4 × 3 = 12.
- Carve off the three right triangles in the box corners: bottom-left (legs 2 and 4) = 4, top-left (legs 1 and 3) = 1.5, top-right (legs 1 and 3) = 1.5.
- Triangle = 12 − (4 + 1.5 + 1.5) = 12 − 7 = 5, so the fraction is 5/30 = 1/6. Good check on the shoelace value.
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