Question

AB and CD are two chords of a circle that $AB=10cm$ and $CD=24cm$ and $AB||CD$. The distance between AB and CD is $17cm$. Calculate the radius of circle.

Answer 1

$AB$ and $CD$ are chords of a circles $D$ and $E$ are mid-points of $AB,CD$ respectively.

$\Rightarrow$ $CF=FD=12cm$ and $AE=EB=5cm$

$\Rightarrow$ Distance between two chords $=17cm$

$\Rightarrow$ Distance between $O$ and $F=xcm$

$\Rightarrow$ Distance between $O$ and $E=(17-x)cm$

In a $△OEB$ is an right angle triangle then,

$\Rightarrow$ $O{B}^2=O{E}^2+E{B}^2$ [ By using Pythagoras theorem ]

$\Rightarrow$ $O{B}^2=(17-x{)}^2+5^2$ -------- ( 1 )

In a $△OFD$ is an right angle triangle then

$\Rightarrow$ $O{D}^2=O{F}^2+F{D}^2$ [ By using Pythagoras theorem ]

$\Rightarrow$ $O{D}^2=(x{)}^2+{12}^2$ ----- ( 2 )

But $OB=OD$ [ Radius of a circle ]

From ( 1 ) and ( 2 )

$\Rightarrow$ $(17-x{)}^2+5^2=(x{)}^2+{12}^2$

$\Rightarrow$ $289+x^2-34x+25=x^2+144$

$\Rightarrow$ $34x=170$

$\therefore$ $x=5$

Substituting value of $x$ in equation ( 2 ) we get,

$\Rightarrow$ $O{D}^2=5^2+{12}^2$

$\Rightarrow$ $O{D}^2=25+144$

$\Rightarrow$ $O{D}^2=169$

$\therefore$ $OD=13cm$

$\therefore$ Radius of a circle is $13cm$