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 AB=6, CD=8 . Suppose further that the area of the part of the circle lying between the chords A

Correct answer: 75 aligned & A=2[(1)/(2) × 25 × θ]+(1)/(2) × 3 × 8+(1)/(2) × 4 × 6 & θ=[π-(θ_1+θ_2)]=[π-( ^ -1 (4)/(3)+ ^ -1 (3)/(4))] & θ=(π)/(2) & A=24+(25 π)/(2) A=(48+25 π)/(2) & (m+n+k)=(48+2+25)=75 aligned

Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Circle

Olympiad workbook · IOQM · Circle

Let A B and C D be two parallel chords in a circle with radius 5...

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

\mathrm{AB}=6, \mathrm{CD}=8

. Suppose further that the area of the part of the circle lying between the chords

A B

and

C D

(m \pi+n) / k

, where

m, n, k

are positive integers with

\operatorname{gcd}(\mathrm{m}, \mathrm{n}, \mathrm{k})=1

. What is the value of

\mathrm{m}+\mathrm{n}+\mathrm{k}

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.

Correct answer

Step-by-step solution

\begin{aligned} & A=2\left[\frac{1}{2} \times 25 \times \theta\right]+\frac{1}{2} \times 3 \times 8+\frac{1}{2} \times 4 \times 6 \\ & \theta=\left[\pi-\left(\theta_1+\theta_2\right)\right]=\left[\pi-\left(\tan ^{-1} \frac{4}{3}+\tan ^{-1} \frac{3}{4}\right)\right] \\ & \theta=\frac{\pi}{2} \\ & A=24+\frac{25 \pi}{2} \Rightarrow \quad A=\frac{48+25 \pi}{2} \\ & (m+n+k)=(48+2+25)=75\end{aligned}

Practice Circle on Quantrex Academy →

More from Circle