Question

Two circles of radius $5cm$ and $4cm$ cut each other at $A$ and $B$. If common chord $AB=6cm$, then find distance between their centers.

Answer 1

Let two circles with center $O$ and $O^{\prime }$ and radius $4cm$ and $5cm$ respectively and $AB$ is common chord.

$OA=5cm,O^{\prime }A=4cm,AB=6cm$

$\therefore AL=\frac{1}{2}AB=\frac{1}{2}\times 6=3cm$

In right angled $\Delta OLA$

$O{A}^2=O{L}^2+A{L}^2$

$(5{)}^2=O{L}^2+(3{)}^2$

$O{L}^2=(5{)}^2–(3{)}^2$

$O{L}^2=25–9$

$OL${2} = 16$

$OL=\sqrt{16}=4$

In right angled $\Delta O^{\prime }LA$

$O^{\prime }{A}^2=O^{\prime }{L}^2+A{L}^2$

$(4{)}^2=O^{\prime }{L}^2+(3{)}^2$

$16=O^{\prime }{L}^2+9$

$O^{\prime }{L}^2=16–9=7$

$O^{\prime }L=\sqrt{7}=2.6457$

$O^{\prime }L=2.65$ (approx)

$O{O}^{\prime }=OL+O^{\prime }L$

$O{O}^{\prime }=4+2.65=6.65cm$