Question

The length of the common chord of two intersecting circles is $30cm$. If the radii of the two circles are $25cm$ and $17cm$, find the distance (in cm) between their centres.

Answer 1

$AB=30cm$

$\therefore AC=BC=\frac{AB}{2}=15cm$

In $\Delta AOC$

$(OA{)}^2=(AC{)}^2+(OC{)}^2$

$OC=\sqrt{O{A}^2-A{C}^2}=\sqrt{{25}^2-{15}^2}$

$OC=\sqrt{(25+15)(25-15)}=\sqrt{40\times 10}=20$

In $\Delta A{O}^{\prime }C$

$(O^{\prime }A{)}^2=(O^{\prime }C{)}^2+(AC{)}^2$

$O^{\prime }C=\sqrt{\left(O^{\prime }A{)}^2\right)-(AC{)}^2}$

$=\sqrt{{17}^2-{15}^2}=\sqrt{(17+15)(17-15)}$

$=\sqrt{32\times 2}=8$

$O{O}^{\prime }=OC+C{O}^{\prime }$

$=20+8=28cm$