Problem 14 · 2016 Math Kangaroo
Hard
Geometry & Measurement
pythagorean-triple
In rectangle ABCD the side BC is exactly half as long as the diagonal AC. Let X be the point on CD for which |AX| = |XC| holds true. How big is the angle ∠CAX?
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Answer: E — another angle
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Hint 1 of 2
Since BC is half the diagonal AC, find the angles the diagonal makes first.
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Hint 2 of 2
Then use the isosceles condition AX = XC; the resulting angle is not one of the first four options.
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Approach: angle chase, then check against the listed values
- BC = AC/2 makes the diagonal split the corner so that angle BAC = 30 degrees and angle ACD = 30 degrees.
- With X on CD and AX = XC, triangle AXC is isosceles, giving angle CAX = 30 degrees.
- That value is not 12.5, 15, 27.5 or 42.5 degrees, so the answer is another angle (E).
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