Problem 30 · 2022 Math Kangaroo
Stretch
Geometry & Measurement
symmetry
Two circles intersect a rectangle AFMG as shown in the diagram. The line segments along the long side of the rectangle that are outside the circles have length AB = 8, CD = 26, EF = 22, GH = 12 and JK = 24. How long is the length x of the line segment LM?
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Answer: C — 16
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Hint 1 of 2
Each circle is symmetric, so the midpoint of the gap it leaves on the top side sits directly above the midpoint of the gap it leaves on the bottom side.
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Hint 2 of 2
Those two alignments, together with the top and bottom sides being equal in length, are enough to solve for x without ever finding a radius.
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Approach: use that each circle's top and bottom gaps share a centre line, plus equal long sides
- Read the top side as AB + arc-gap + CD + middle-gap + EF = 8 + … + 26 + … + 22, and the bottom as GH + … + JK + … + x = 12 + … + 24 + … + x.
- Because each circle is symmetric, the midpoint of its top chord lies exactly above the midpoint of its bottom chord; the two alignment conditions force (top chord − bottom chord) of circle 1 to be 8 and the corresponding middle-gap difference to be −12.
- Setting the top side equal to the bottom side gives x = 20 + 8 − 12 = 16.
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