Question

In the following figure,what will be the value of (AD)?

Solution:
We are given that the perimeter's of triangleADB and triangleADC are equal.

The first step would be to find the length of BD and DC.

Since the perimiters are equal we get AB+BD+AD=AD+DC+CA

=> AB+BD=AC+DC

=> 60+100-DC=80+DC

=> DC=40 so BD = 60

Now there are two ways to solve this question. First is a simple one which requires some creativity or construction and basic knowledge of parallel lines.

Let us draw a line PD parallel to AC, so we will get two right angled triangles BPD and PDA. Since PD is parallel to AC

((BD)/(BC))=((BP)/(BA))

That is, (60/100)=((BP)/60)
Therefore, BP = 36

Now in triangle BDP,applying pythagoreas theorem, we get,

(PD^2) = (BD^2)-(BP^2)

PD=sqrt[60^2) - (36^2)]

PD=48

Again, in triangle PDA,applying pythagoreas theorem, we get,
(AD^2)=PD^2+AP^2

AD=sqrt[(48^2) + (24^2)]

AD=sqrt(2880)

Therefore, AD=24sqrt(5)

Alternatively a more mudane approach would be to look at the triangle ABD. Since we know two sides and an angle we should be able to apply the formula.

AD^2=BD^2+AB^2-2BDxxABcosABD

Solving this we will get AD = 24sqrt(5)