Problem 10 · 2016 Math Kangaroo
Medium
Geometry & Measurement
careful-counting
The diagram shows a circle with centre O as well as a tangent that touches the circle at point P. The arc AP has length 20 and the arc BP has length 16. What is the size of the angle ∠AXP?
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Answer: E — 10°
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Hint 1 of 3
Notice A, O, B are collinear, so AB is a diameter and the arc from A through P to B is a semicircle.
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Hint 2 of 3
The line through X is tangent at P and a secant cutting A and B, so its angle equals half the difference of the two intercepted arcs.
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Hint 3 of 3
Turn the arc lengths into degrees first, using that the two arcs add to 180.
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Approach: convert arc lengths to degrees, then use the tangent-secant angle
- Since A, O, B lie on one line, AB is a diameter, so arc AP + arc PB is a semicircle: \(20 + 16 = 36\) units of length equal \(180^\circ\), i.e. 1 unit is \(5^\circ\).
- Thus arc AP \(= 100^\circ\) and arc PB \(= 80^\circ\).
- The tangent at P and the secant through B and A meet at X, so \(\angle AXP = \tfrac12(\text{arc }AP - \text{arc }PB) = \tfrac12(100^\circ - 80^\circ) = 10^\circ\).
- So the angle is 10° (E).
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