Draw two concentric circles of radii 3 cm and 5 cm. Construct a tangent to smaller circle from a point on the larger circle. Also measure its length.

#### Solution

Following are the steps to draw tangents on the given circle:

**Step 1**

Draw a circle of 3 cm radius with centre O on the given plane.

**Step 2**

Draw a circle of 5 cm radius, taking O as its centre. Locate a point P on this circle and join OP.

**Step 3**

Bisect OP. Let M be the midpoint of PO.

**Step 4**

Taking M as its centre and MO as its radius, draw a circle. Let it intersect the given circle at points Q and R.

**Step 5**

Join PQ and PR. PQ and PR are the required tangents.

It can be observed that PQ and PR are of length 4 cm each.

In ΔPQO,

Since PQ is a tangent,

∠PQO = 90°

PO = 5 cm

QO = 3 cm

Applying Pythagoras theorem in ΔPQO, we obtain

PQ^{2} + QO^{2} = PQ^{2}

PQ^{2} + (3)^{2} = (5)^{2}

PQ^{2} + 9 = 25

PQ^{2 }= 25 − 9

PQ^{2 }= 16

PQ = 4 cm

Hence justified.