In the given fig, bisectors of ∠B and ∠C of ∆ABC intersect each other in point X. Line AX intersects side BC in point Y. AB = 5, AC = 4, BC = 6 then find AX / AY

Answer:-

First, let’s summarize the given information and what we need to find:

• AY is the bisector of angle A.
• CX is the bisector of angle CBY.
• AB/AC = 5/4, BC/AC = 9/4.
• We need to find AX/XY.

Using the angle bisector theorem in triangle ACY, we have:

AX/XY = AC/YC = 4/YC ……(1)

Using the angle bisector theorem in triangle CBY, we have:

BY/YC = AB/AC = 5/4

BC/YC = 9/4 – (BY/YC) = 1/4

Now, we can find YC:

YC/BC = 4/1

YC = (4/1)BC = (4/1)(9/4)AC = 9/3 = 3

Therefore, from equation (1), we have:

AX/XY = 4/YC = 4/3

Since AY is the bisector of angle A, we have:

AX/XY = AB/YB

Using the fact that AB/AC = 5/4 and BC/YC = 1/4, we can write:

AB/YB = AB/(AY + YB) = AB/(AC + BC) = 5/13

Therefore, we have:

4/3 = 5/13

Cross-multiplying gives:

39 = 15XY

So, XY = 39/15 = 2.6.

Hence, AX/XY = (4/3) = 1.3.