Question

Two circles of radii $10$ cm and $17$ cm intersecting each other at two points and the distance between their centres is $21$ cm. Find the length of the common chord.

Answer 1

Given two circle A & B of radii $10$cm of $17$cm. The distance between their centres is $21$cm. Let $P\theta$ be common tangent to circle A & B.

$PB=17cm$ & $AP=10cm$

Let $BM=x$ then $AM=21-x$ & $PM=y$ then using Pythagoras is $\Delta PBM$

$(PB{)}^2=(PM{)}^2+(BM{)}^2$

$x^2+y^2=289$ …………$\left(1\right)$

In $\Delta APM$ using Pythagoras theorem

$(AP{)}^2=(AM{)}^2+(PM{)}^2$

$100=x^2+(21-y{)}^2$

$x^2+y^2+441-42y=100$

From equation $\left(1\right)$

$289+441-100=42y$

$42y=730-10-630$

$y=\frac{630}{42}$

$y=15$

Now length of chord $=2y=30$

[Length of common chord $=30cm$].