# A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD + BC

**Solution:**

From the given figure we can consider a few points which are as follows:

(i) DR = DS [ As we know that the lengths of the tangents drawn from an external point to the circle are equal.]

(ii) BP = BQ

(iii) AP = AS

(iv) CR = CQ

Since they are tangents on the circle from points D, B, A, and C respectively.

Now let us add both the LHS and RHS of the above equations separately and observing the result.

DR + BP + AP + CR = DS + BQ + AS + CQ

By rearranging the terms we get,

(DR + CR) + (BP + AP) = (CQ + BQ) + (DS + AS)

On further simplifying,

CD + AB = BC + AD

Hence it is proved AB + CD = AD + BC.

**Video Solution:**

## A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD + BC

### NCERT Solutions Class 10 Maths - Chapter 10 Exercise 10.2 Question 8:

**Summary:**

If a quadrilateral ABCD is drawn to circumscribe a circle, then AB + CD = AD + BC.