 Hi and welcome to the session. Let's work out the following question. The question says D and E are points on the sides AB and AC respectively of triangle ABC such that DE is parallel to BC and AD is to DB is equal to 4 is to 5. CD and BE intersect each other at F, find the ratio of the areas of triangle DEF and triangle BCF. So let us see the solution to this question. Now in the question we are given triangle ABC such that DE is parallel to BC and AD is to DB is in the ratio 4 is to 5. Let us call this 4k so this will be 5k. Now what we are supposed to find in this question is area of triangle DEF divided by area of triangle BCF. Let us start with the proof now. We see that AD is to DB is equal to 4 upon 5 that is given to us in the question. So we can say that AD is equal to 4k and DB is equal to 5k. This is what we have named them here. Now in triangle ADE and triangle ABC, angle 1 is equal to angle 1 that is a common angle in both the triangles. Angle 2 is equal to angle ABC because they are the corresponding angles for the parallel lines DE and BC where AV acts as a transversal. Therefore triangle ADE is similar to triangle ABC by angle angle similarity criterion since these two are similar triangles therefore AD upon AB is equal to AE upon AC is equal to DE upon BC because in similar triangles corresponding sides are proportional. This implies 4k upon 9k because AB is 4k plus 5k that is 9k is equal to DE upon BC. This implies that DE upon BC is equal to 4 upon 9 and we call this equation 1. Now in triangle DEF and triangle BCF that is this triangle and this triangle angle 3 is equal to angle 6 because they are the alternate interior angles. Secondly angle 4 is equal to angle 5 because they are the vertically opposite angles that is this angle is equal to this angle therefore triangle DEF is similar to triangle BCF again by angle angle similarity criterion. Since these two triangles are similar therefore area of triangle DEF upon area of triangle BCF is equal to DE square by BC square because we see that areas of two triangles are in the ratio of the squares of the corresponding sides. So this will be equal to 4k upon 9k the whole square this we get by this equation and this is equal to 16 by 81. So our answer to this question is that ratio of areas of triangle DEF and triangle BCF is 16 upon 81. So this is what we will we have to find this question. I hope that you understood the solution and enjoyed the session. Have a good day.