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 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’ be the mid-point of OP. Taking M’ as centre and OM’ as radius draw a circle dotted which cuts the inner circle at M and P’.
4. Join PM and PP’. Thus, PM and PP’ are the required tangents.
5. On measuring PM and PP’, we find that PM = PP’ = 4 cm.

Actual calculation:

In right angle ∆OMP,

∠PMO = 90°

PM² = OP² – OM²  ...[By Pythagoras theorem i.e. (hypotenuse)² = (base)² + (perpendicular)²]

⇒ PM² = (5)² – (3)²

= 25 – 9

= 16

⇒ PM = 4 cm

Hence, the length of both tangents is 4 cm.