Problem 8 · 2019 Math Kangaroo
Medium
Geometry & Measurement
spatial-reasoning
Which of the following statements is definitely true for the angle marked in the diagram, which is made up of nine squares?
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Answer: B — 2α + β = 90°
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Hint 1 of 2
Put the picture on a grid; each slanted line goes from the corner to a lattice point.
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Hint 2 of 2
The two lines have slopes 2/3 and 3/2 — reciprocals. Look for a clean angle relation.
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Approach: read the slopes and combine the angles
- From the corner, the lower line has slope 2/3 and the upper line slope 3/2, so α = arctan(2/3) and α + β = arctan(3/2).
- Since 2/3 and 3/2 are reciprocals, arctan(2/3) + arctan(3/2) = 90°.
- That sum equals α + (α + β) = 2α + β, so 2α + β = 90°.
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