Problem 17 · 2020 Math Kangaroo
Stretch
Geometry & Measurement
symmetryspatial-reasoning
Two circles are tangent to each other and also to two sides of a square. What is the measure of the angle AÔB, determined by three of these points of tangency, as shown in the figure?
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Answer: E — 135°
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Hint 1 of 2
The two equal circles and their three tangent points sit symmetrically about a diagonal of the square.
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Hint 2 of 2
Drop the radii to the tangent points (each radius meets its tangent line at a right angle) and chase the angles.
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Approach: use radii-perpendicular-to-tangents plus the diagonal symmetry
- The two circles are equal and lined up along the square’s diagonal, with O the point where they touch each other.
- A radius meets each tangent line at 90°, so the angle from O to each outer tangency point is built from a right angle plus the 45° of the diagonal.
- Adding the two symmetric halves gives the angle AÔB = 135°, choice E.
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