# Draw two concentric circles of radii 3 cm and 5 cm. Taking a point on outer circle construct the pair of tangents to the other. Measure the length of a tangent and verify it by actual calculation.

Solution

Steps of construction
1.   Draw two concentric circles with center O and radii 3 cm and 5 cm
2.   Taking any point P on outer circle, Join P and O
3.   Draw perpendicular bisector of OP let M be the mid point of OP
4.   Taking M as centre and OM as radius draw a circle which cuts inner circle at Q and R
5.   Join PQ and PR. Thus PQ and PR are required tangents
On measuring PQ and PR we find that PQ = PR = 4 cm
Calculations
$\bigtriangleup OQP, \angle OQP= 90^{\circ}$
$OP^{2}= OQ^{2}+PQ^{2}\left$ [using pythagoras theorem]
$\left ( 5 \right )^{2}= \left ( 3 \right )^{2}+\left ( PQ \right )^{2}$
$25-9= PQ^{2}$
$16= PQ^{2}$
$\sqrt{16}= PQ$
$4cm= PQ$
Hence the length of both tangents is 4 cm.

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