Question

Find the length of a tangent to a circle of radius $4$ cm, from a point on the concentric circle of radius $6$ cm.

• A. $\sqrt{20}$ cm
• B. $\sqrt{10}$ cm
• C. $\sqrt{5}$ cm
• D. $\sqrt{30}$ cm

Answer 1

The correct option is A $\sqrt{20}$ cm

Steps of construction :
Step 1 : Draw two concentric (same centre) circles with radii 4 cm. and 6 cm. respectively and centre O.
Step 2 : Take a point P on outer circle and join OP.
Step 3 : Bisect OP and let the mid-point of OP be M.
Step 4 : Draw a circle taking M as centre and radius equal to MP intersecting the smaller circle at Q and R.
Step 5 : Join PQ and PR.
Thus PQ and PR are tangents to the circle (O, 4 cm.). On measuring PQ and PR we find that $PQ=PR=\sqrt{20}cm$
Since $\angle$PO is in semicircle
$\therefore \angle PQO={90}^{\circ }$
Now in right triangle PQO
$O{P}^2=O{Q}^2+P{Q}^2$
$\Rightarrow 6^2=4^2+P{Q}^2$
$\Rightarrow PQ=\sqrt{36-16}=\sqrt{20}cm$