Question

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.

Answer 1

Given, two concentric circles of radii 3 cm and 5 cm with centre O, we have to draw a pair of tangents from point P on outer circle to the other.
Steps of construction
1. Draw two concentric circles with centre O and radii 3 cm and 5 cm,

2. Taking any point P on outer circle . Join OP

3. Bisect OP. Let $M^{{}^{\prime }}$ be the mid-point of OP
Taking $M^{{}^{\prime }}$ as centre and $O{M}^{{}^{\prime }}$ as radius draw a circle dotted which cuts the inner circle at M and $P^{{}^{\prime }}$

4. Join PM and $P{P}^{{}^{\prime }}$ Thus, PM and $P{P}^{{}^{\prime }}$ are the required tangents.

5. On measuring PM and $P{P}^{{}^{\prime }}$, we find that $PM=P{P}^{{}^{\prime }}=4cm$.

Actual calculation

In right angle $\Delta OMP$ $\angle PMO={90}^{\circ }$

$\Rightarrow P{M}^2=O{P}^2-O{M}^2$

[ by Pythagoras theorem i.e $(hypotenuse{)}^2$ =$(base{)}^2$ + $(perpendicular{)}^2$]

$\Rightarrow P{M}^2=O{P}^2-O{M}^2$

$\Rightarrow PM=4cm$

Hence, the length of both tangents is 4 cm.