# In the given figure, a circle with centre O, is inscribed in a quadrilateral ABCD such that it touches the side BC,

**Question:**

In the given figure, a circle with centre O, is inscribed in a quadrilateral ABCD such that it touches the side BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29 cm, AD = 23 cm, ∠B = 90∘ and DS = 5 cm then find the radius of the circle.

**Solution:**

We know that tangent segments to a circle from the same external point are congruent.

Now, we have

DS = DR, AR = AQ

Now, AD = 23 cm

⇒ AR + RD = 23

⇒ AR = 23 − RD

⇒ AR = 23 − 5 [∵ DS = DR = 5]

⇒ AR = 18 cm

Again, AB = 29 cm

⇒ AQ + QB = 29

⇒ QB = 29 − AQ

⇒ QB = 29 − 18 [∵ AR = AQ = 18]

⇒ QB = 11 cm

Since all the angles are in a quadrilateral BQOP are right angles and OP = BQ.

Hence, BQOP is a square.

We know that all the sides of square are equal.

Therefore, BQ = PO = 11 cm

Hence, the radius of the circle is 11 cm.