In a `Delta ABC, AB = 52, BC = 56, CA = 60`. Let D be the foot of the altitude from `A and E` be the intersection of the internal angle bisector of `/_BAC` with BC. Find the length DE.

Answer

6

Explanation

Let AD = h
from the figure,
In `DeltaABD, h^(2) + x^(2) = (52)^(2)`
In `DeltaADC, h^(2) + (56 -x)^(2) = (60)^(2)`
Solving, we get x = 20 and h = 48
Let BE = y. Then from the property of angle bisector, we have
`(AB)/(AC) = (BE)/(EC)`
or `(52)/(60) = (y)/(56 -y)`
or `60y = 52 xx 56 - 52 y`
or `112y = 52 xx 56`
or `y = 26`
`rArr DE = BE - BD`
`= (26 - 20)`
`= 6`