Problem 7 · 2009 AMC 8
Easy
Geometry & Measurement
triangle-area
The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD?
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Answer: C — 4.5 square miles.
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Hint 1 of 2
The triangle looks slanted, but you get to CHOOSE which side is the base. Pick the side CD sitting on the (vertical) railroad — then the height is just the straight horizontal distance over to A.
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Hint 2 of 2
Base and height must be perpendicular. The railroad runs north–south and Main Street runs east–west, so they're already at right angles — that's why those two segments are the easy base/height pair.
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Approach: choose a convenient base, read its perpendicular height
- Use CD as the base: C and D both lie on the railroad, with CD = 3.
- The height is the perpendicular distance from A across to that railroad line. Since Main Street (east–west) meets the railroad (north–south) at a right angle, that distance is just AB = 3.
- Area = ½ × base × height = ½ × 3 × 3 = 4.5.
- Why this transfers: on a tilted triangle, find a side that lies along a grid line or axis — its perpendicular height is then a simple horizontal or vertical gap, sparing you any slope or distance-formula work.
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