Super combination

Geometry Level 2

O X OX , O Y OY are radii of a circular quadrant. A semi-circle is drawn on X Y XY as shown. T , S T, S and C C denote the resulting triangle, segment and crescent.

Find the ratio of

Area T Area C \large \dfrac{\text{Area } T}{\text{Area } C}

15 4 π \dfrac{15}{4 \pi} 7 2 π \dfrac{7}{2 \pi} 1 3 π \dfrac{3}{\pi} 13 4 π \dfrac{13}{4 \pi}

Syed Hamza Khalid by Syed Hamza Khalid , 17, France

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1 solution

Jeremy Galvagni
Aug 13, 2018

Let O X = 1 OX=1 then Area T = 1 2 T=\frac{1}{2} , Area S = π 4 − 1 2 S=\frac{\pi}{4}-\frac{1}{2}

The radius of the semicircle is 2 2 \frac{\sqrt{2}}{2} so the area of S + C = π 4 S+C=\frac{\pi}{4}

Subtract the area of S from this to get Area C = 1 2 C=\frac{1}{2}

So T and C have the same area and the ratio is 1 \boxed{1}

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