Question

Construct a tangent to a circle of radius $4cm$ from a point on the concentric circle of radius $6cm$ and measure its length. Also, verify the measurement by actual calculation.

Answer 1

Steps of construction

1) Draw a circle of radius $4cm$ from the centre $O$. With same centre and radius equal to $6cm$, construct another circle.

2) Take any point $P$ on the circumference of the outer circle and join $OP$.

3) Construct perpendicular bisector for the line segment $PO$, which intersect $OP$ at point $M$

4)Now, with $M$ as centre, construct a circle of radius equal to $PM$ or $MO$.

5) This circle now intersects the inner circle at points $Q$ and $R$. Join $PQ$ and $PR$.

6) Thus, tangents have been constructed from outer circle to the inner circle.

7) On measuring $PR$ or $PQ$, we get $PR=PR=4.4cm$ (approx)

Verification

$PO$ acts as a diameter for the smallest circle and hence,

$\angle PQO=\angle PRO={90}^{\circ }$ [Angle in the semi circle]

Thus, $OQ\perp PQ$ and $OR\perp PR$ hence, $PQ$ and $PR$ are tangents.

Consider, right $\Delta PQO$

$\Rightarrow P{O}^2=P{Q}^2+O{Q}^2$

$\Rightarrow 6^2=P{Q}^2+4^2$

$\therefore PQ=\sqrt{36-16}=\sqrt{20}=\mathrm{4..47}cm$ (approx)

Hence, verified.