Question

The lengths of three consecutive sides of a quadrilateral circumscribing a circle are 4 cm, 5 cm and 7 cm respectively. Determine the length of the fourth side.

Answer 1

Let us consider a quadrilateral ABCD, AB = 4cm, BC = 5 cm, CD = 7cm, CD as sides

circumscribing circle with centre O. and intersecting at points E, F, G, H. as in fig.

We know that the tangents drawn from external point to the circle are equal in length.

From point A, AE = AH …. (i)

From point B, BE = BF ….. (ii)

From point C, GC = CE ….(iii)

From point D, GD = DH ….(iv)

(i) + (ii) + (iii) + (iv) ⇒ (AE + BE + GC + GD) = (AH + BF + CF + DH)

⇒ (AE + BE) + (GC + GD) = (AH + DH) + (BF + CF)

⇒ AB + CD = AD + BC

⇒ AD = 11 – 5 = 6 cm

Fourth side = 6 cm