Points P and Q lie on sides AB and AC of triangle ABC respectively such that segment PQ is parallel to side BC. If the ratio of areas of ΔAPQ:ΔABC is 25:36, then the ratio of AP:PB is:

Since PQ∥BC, ΔAPQ is similar to ΔABC (ΔAPQ∼ΔABC) by AA similarity criterion.
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides:

Area(ΔABC)
Area(ΔAPQ)
​
=(
AB
AP
​
)
2

Given:

Area(ΔABC)
Area(ΔAPQ)
​
=
36
25
​

Taking the square root on both sides:

AB
AP
​
=
36

​

25

​

​
=
6
5
​

We know that AB=AP+PB. Therefore:

AP+PB
AP
​
=
6
5
​

6AP=5(AP+PB)
6AP=5AP+5PB
6AP−5AP=5PB
AP=5PB
PB
AP
​
=
1
5
​

Thus, the ratio AP:PB=5:1.