Note:If centre of the circle is not given, (Ex 11.2, 7)

We find its center first by

- Taking any two non-parallel chords
- And then finding the point of intersection of their perpendicular bisectors.
Then you could proceed as

Construction 11.3.

**-v-**

**Construction 11.3:**

To construct the tangents to a circle from a point outside it.

We are given a circle with centre O and a point P outside it. We have to construct the two tangents from P to the circle.

**Steps ****of Construction****:**

- Join PO and bisect it. Let M be the midpoint of PO.
- Taking M as centre and MO as radius, draw a circle.
- Let it intersect the given circle at the points Q and R.
- Join PQ and PR.

Then, PQ and PR are the required two tangents.

**Justification:**

__We ____need to prove that PQ and PR are the tangents to the circle.__

Join OQ and OR.

∠PQO is an angle in the semi-circle

of the blue circle

And we know that angle in a

semi-circle is a right angle.

∴ ∠PQO = 90°

⇒ OQ ⊥ PQ

Since OQ is the radius of the circle,

PQ has to be a tangent of the circle.

Similarly, PR is a tangent of the circle.