Try this beautiful Problem based on Chords in a Circle, Geometry, from PRMO 2017.
Let $A B$ and $C D$ be two parallel chords in a circle with radius 5 such that the centre $O$ lies between these chords. Suppose $A B=6, C D=8 .$ Suppose further that the area of the part of the circle lying between the chords $A B$ and $C D$ is $(m \pi+n) / k,$ where $m, n, k$ are positive integers with gcd$(m, n, k)=1$ . What is the value of $m+n+k ?$
Geometry
Triangle
Circle
Answer:$75$
PRMO-2017, Problem 26
Pre College Mathematics
Draw OE $\perp A B$ and $O F \perp C D$
Clearly $\mathrm{EB}=\frac{\mathrm{AB}}{2}=3, \mathrm{FD}=\frac{\mathrm{CU}}{2}=4$
$\mathrm{OE}=\sqrt{5^{2}-3^{2}}=4$ and $\mathrm{OF}=\sqrt{5^{2}-4^{2}}=3$
Therefore $\Delta O E B \sim \Delta D F O$
Can you now finish the problem ..........
Let $\angle \mathrm{EOB}=\angle \mathrm{ODF}=\theta,$ then
$\angle B O D=\angle A O C=180^{\circ}-\left(\theta+90^{\circ}-\theta\right)=90^{\circ}$
Now area of portion between the chords
= \(2 \times\) (area of minor sector BOD)+2 \times ar\((\triangle AOB)\)
$=2 \times \frac{\pi \times 5^{2}}{4}+2 \times \frac{1}{2} \times 6 \times 4=\frac{25 \pi}{2}+24=\frac{25 \pi+48}{2}$
Therefore $m=25, n=48$ and $k=2$
Can you finish the problem........
Therefore $m+n+k=75$
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