Question

A circle touches all the sides of ؈ABCD. If AB is the largest side then prove that CD is the smallest side.

Answer 1

A circle touches all the four sides of quadirateral ABCD and also it is given that AB is the largest side .

we have to prove that CD is the smallest sides.

proof :- we know that if circle touches all sides of ABCD
then,  AB + CD = BC + DA …......(i)
Given AB is the largest side.
⇒ AB > BC
Let m, in such a way that AB = BC + m
now from eq. (i),
⇒ BC + m + CD = BC + DA
⇒ CD + m = DA
it seems that CD is smaller than DA or we can say that DA is greater than CD.
∴ CD < DA
Hence CD is smaller than DA. …....(ii)
But AB is the largest side.
⇒ AB > DA
again , we let n, in such a way that AB = DA + n
From eq. (i),
⇒ DA + n + CD = BC + DA
⇒ CD + n = BC
∴ CD < BC
Hence CD is smaller than BC. … (iii)
according to question,
AB is largest side, so CD is smaller than AB....... (4)
From (ii), (iii) and (iv),
it is clear that CD is the smallest side of ABCD.