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Try this beautiful problem from the Pre-RMO, 2017, Question 26, based on Area of part of circle.

## Area of part of circle – Problem 26

Let AB and CD be two parallel chords in a circle with radius 6 such that the centre O lies between these chords. Suppose AB=6 and CD=8. Suppose further that the area of the part of the circle lying between the chords AB and CD is $\frac{m\pi+n}{k}$ where m.n.k are positive integers with gcd(m,n,k)=1. What is the value of m+n+k?

• is 107
• is 75
• is 840
• cannot be determined from the given information

### Key Concepts

Equation

Algebra

Integers

But try the problem first…

Source

PRMO, 2017, Question 26

Higher Algebra by Hall and Knight

## Try with Hints

First hint

A=2[$\frac{1}{2} \times 25 \times \theta$]+$\frac{1}{2} \times 3 \times 8$+$\frac{1}{2} \times 4 \times 6$

where $\theta=[\pi-(\theta_1+\theta_2)]=[\pi-(tan^{-1}\frac{4}{3}+tan^{-1}\frac{3}{4})]$

Second Hint

or, $\theta=\frac{\pi}{2}$

or, A=24+$\frac{25\pi}{2}$

or, A=$\frac{48+25\pi}{2}$

Final Step

(m+n+k)=(48+2+25)=75