Problem 15 · 2011 Math Kangaroo
Hard
Geometry & Measurement
A car's rear window wiper is built so that the rod r and the wiper blade w are equally long and are joined at an angle α. The wiper turns about the centre of rotation O and wipes the area shown on the right. Find the angle β between the right edge of the cleaned area and the tangent to the curved upper edge.
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Answer: B — \(\pi-\frac{\alpha}{2}\)
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Hint 1 of 2
The blade tip traces a circle about O, and a tangent to that circle is perpendicular to the radius.
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Hint 2 of 2
Triangle O–joint–tip is isosceles (rod = blade), so its base angle is (π−α)/2.
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Approach: use the isosceles rod-blade triangle plus the tangent-radius right angle
- Rod and blade are equal, so the triangle from O to the joint to the tip is isosceles with apex angle α; its base angle is (π−α)/2.
- The tangent to the swept arc is perpendicular to the radius to the tip.
- Adding the right angle to the base angle: β = π/2 + (π−α)/2 = π − α/2.
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