Question

What is the distance between two parallel tangents of a circle having radius $4.5cm$ ? Justify you answer. [3 Marks]

Answer 1

Let the lines $PQ$ and $RS$ be the two parallel tangents to circle at $M$ and $N$ respectively.
Through centre $O$, draw line $AB\parallel$ line RS.

$OM=ON=4.5$ [Given]
line $AB\parallel$ line $RS$ [Construction]
line $PQ\parallel$ line $RS[$ Given]

$\therefore$ line $AB\parallel$ line $PQ\parallel$ line $RS$ Now, $\angle OMP=\angle ONR={90}^{\circ }$ (i) [Tangent theorem]
For line $PQ\parallel$ line $AB$,

$\angle OMP=\angle AON={90}^{\circ }$ (ii) [Corresponding angles and from (i)] [1 Mark]

For line $RS\parallel$ line $AB$,
$\angle ONR=\angle AOM={90}^{\circ }$ (iii) [Corresponding angles and from (i)]

$\angle AON+\angle AOM={90}^{\circ }+{90}^{\circ }[$ From (ii) and (iii)] $\therefore \angle AON+\angle AOM={180}^{\circ }$ [1 Mark]

$\therefore \angle AON$ and $\angle AOM$ form a linear pair.
$\therefore$ ray OM and ray ON are opposite rays.
$\therefore$ Points $M,O,N$ are collinear. (iv) $\therefore MN=OM+ON[M-O-N$, From (iv)] $\therefore MN=4.5+4.5$
$\therefore MN=9cm$
$\therefore$ Distance between two parallel tangents $PQ$ and $RS$ is $9cm.$
[1 Mark]