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Problem of the Week

Problem E

Cupid’s Arrow

At a Valentine’s dance, contestants can participate in a game. They are blindfolded and spun around. They then try to place an arrow on a white heart on a red gameboard. (See the diagram below.) The heart was constructed by attaching two white semi-circles onto the hypotenuse of an isosceles right-angled triangle. Each semi-circle has the same diameter, equal to half the length of the hypotenuse. The heart was then pasted onto a large rectangular sheet of red paper such that the hypotenuse of the triangle is parallel to the base of the rectangle. We know the following measurements:

(The dashed lines, the length measurements and the right angle symbol will not actually be on the finished gameboard.) If a contestant places their arrow randomly somewhere on the gameboard, what is the probability that it will land on the white heart?

The rectangular gameboard with a white heart. A dashed horizontal line divides the heart along the hypotenuse of the right-angled triangle which forms the bottom of the heart. The right angle is at the bottom point of the heart. The various length measurements are shown on the gameboard.

Note: To solve the problem, it may be helpful to use the following fact:
In an isosceles triangle, the line joining the vertex opposite the unequal side to the midpoint of the unequal side will be perpendicular to the unequal side.