Problem 20 · 2016 Math Kangaroo
Hard
Geometry & Measurement
careful-counting
Bettina chooses five points A, B, C, D and E on a circle and draws the tangent to the circle at point A. She realizes that the five angles marked x are all equally big. (Note that the diagram is not drawn to scale!) How big is the angle ∠ABD?
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Answer: C — 72°
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Hint 1 of 3
The tangent at A and the chords AB, AC, AD, AE split the straight tangent line into the five equal angles x.
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Hint 2 of 3
A tangent-chord angle equals half its intercepted arc, so each angle x cuts off an equal arc.
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Hint 3 of 3
Add up the equal arc pieces that the inscribed angle ABD intercepts.
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Approach: equal tangent-chord angles cut equal arcs
- The five equal angles x fill the straight tangent line at A, so \(5x = 180^\circ\) and \(x = 36^\circ\).
- Each tangent-chord step cuts an arc of \(2x = 72^\circ\), so going A, B, C, D the chord AD reaches \(3 \times 72^\circ = 216^\circ\) around from A.
- The inscribed angle ABD intercepts arc AD on the side not containing B, namely \(360^\circ - 216^\circ = 144^\circ\).
- So \(\angle ABD = \tfrac12 \times 144^\circ = 72^\circ\), answer C.
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