 Hello and welcome to the session. Let us discuss the following question. It says d and e are points on the sides a, b and a, c respectively of triangle a, b, c, so that d e is parallel to b, c. And d a, d is to d, b is equal to 4 is to 5 and c, d and b, e intersect each other at f. Find the ratio of the area of the triangle, d e f and b, c f. So this is the triangle a, b, c in which d e is parallel to b, c. We have to find the ratio of the areas of the triangle d e f and b, c f. So let us now move on to the solution. Now we are given that a, d is to d, b is equal to 4 is to 5. So a, d is equal to 4 k and d b is equal to 5 k where 5 k is the constant. Now since a, b is equal to a, d plus d b. So a, b is equal to 9 k. Now in triangle a, d e and a, b, c we have angle a is equal to angle a since it is a common angle. Angle a, d e is equal to angle a, b, c. That is this angle is equal to this angle. This is because d e is parallel to b, c and these are corresponding angles. Therefore triangle a, d e is similar to triangle a, b, c by angle, angle similarity. Now since triangle a, d e is similar to a, b, c therefore corresponding sides are proportional. So we have a, d upon a, b is equal to a, e upon a, c is equal to d e upon b, c. Now a, d upon a, b is equal to 4 k upon 9 k, a, d is 4 k and a, b is 9 k which is equal to a, e upon a, c is equal to d e upon b, c. So we have d e upon b, c is equal to 4 by 9. Now in triangle d e, f and triangle b, c, f angle e, d, f is equal to angle b, c, f. So these are alternate angles as this angle is equal to this angle. Angle is equal to angle f, c, b, alternate angles. Angle b, f, e is equal to angle b, f, c and these are vertically opposite angle again by angle, angle similarity. Triangle b, e, f is similar to triangle b, c, f, angle, angle similarity. Therefore area of triangle b, e, f upon area of triangle b, c, f is equal to d e square upon b, c square. As we know that the areas of two similar triangles are in the ratio of the areas of the corresponding sides. Similar triangles are in the ratio of the squares of corresponding side. Now d e upon b, c square is equal to 4 by 9 square that is 16 by 81. This is by using one. Hence area of triangle d e, f upon area of triangle b, c, f is equal to 16 by 81. So this is the required ratio. This completes the question and the session. Bye for now. Take care and have a good day.