Try this beautiful problem from the Pre-RMO, 2017, Question 26, based on Area of part of circle.
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?
Equation
Algebra
Integers
But try the problem first...
Answer: is 75.
PRMO, 2017, Question 26
Higher Algebra by Hall and Knight
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
Try this beautiful problem from the Pre-RMO, 2017, Question 26, based on Area of part of circle.
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?
Equation
Algebra
Integers
But try the problem first...
Answer: is 75.
PRMO, 2017, Question 26
Higher Algebra by Hall and Knight
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
