Question

In a circle of radius $5cm,AB$ and $CD$ are two parallel chords of lengths $8cm$ and $6cm$ respectively.
Calculate the distance between the chords if they are on opposite sides of the centre.

Answer 1

Consider $AB$ and $CD$ as the chords on the circle and $AB\parallel CD$ on the opposite sides of the centre
It is given that $AB=8cm$ and $CD=6cm$
Construct $OL\perp AB$ and $OM\perp CD$
Join the diagonals $OA$ and $OC$
We know that $OA=OC=5cm$
Perpendicular from the centre of a circle to a chord bisects the chord
We know that $AL=1/2\times AB$
By substituting the values we get
$AL=1/2\times 8$
So we get
$AL=4cm$
We know that $CM=1/2\times CD$
By substituting the values we get
$CM=1/2\times 6$
So we get
$CM=3cm$
Consider $△OLA$
Using the Pythagorean theorem it can be written as
$O{A}^2=A{L}^2+O{L}^2$
By substituting the values
$5^2=4^2+O{L}^2$
On further calculation
$O{L}^2=25-16$
By subtraction
$O{L}^2=9$
By taking the square root
$OL=\surd 9$
$OL=3cm$
Consider $△OMC$
Using the Pythagorean theorem it can be written as
$O{C}^2=O{M}^2+C{M}^2$
By substituting the values
$5^2=O{M}^2+3^2$
On further calculation
$O{M}^2=5^2-3^2$
$O{M}^2=25-9$
By subtraction
$O{M}^2=16$
By taking the square root
$OM=\surd 16$
$OM=4cm$
Distance between the chords $=OM+OL$
So we get
Distance between the chords $=4+3=7cm$
Therefore, the distance between the chords on the opposite sides of the centre is $7cm$