Problem 21 · 2019 AMC 8
Medium
Geometry & Measurement
area
What is the area of the triangle formed by the lines y = 5, y = 1 + x, and y = 1 − x?
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Answer: E — 16.
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Hint 1 of 2
Before computing, notice the two slanted lines are mirror images: slopes +1 and −1, both crossing at the same apex. That symmetry makes the horizontal line y = 5 a perfect flat base.
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Hint 2 of 2
Pick the horizontal side as the base — its length is just the gap between two x-values at y = 5, and the height is the straight vertical drop to the apex. No slant distances needed.
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Approach: use the horizontal line as base; exploit the ±1 symmetry
- Find the corners. On y = 5: 5 = 1 + x gives (4, 5), and 5 = 1 − x gives (−4, 5). The two slanted lines meet where 1 + x = 1 − x, i.e. x = 0, the apex (0, 1).
- Take the flat top as base: from (−4, 5) to (4, 5) is length 8 (and it's centered on the y-axis, confirming the triangle is isosceles). Height is the vertical distance 5 − 1 = 4.
- Area = ½ × 8 × 4 = 16.
- Why this transfers: when a triangle has a horizontal or vertical side, use that side as the base — its length and the perpendicular height are both just coordinate differences, sidestepping the distance formula entirely.
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