 Hello students, let's work out the following problem. It says, Diagnose of a trapezium A, B, C, D with A, B parallel to D, C intersect each other at O. If A, B is twice of C, D, find the ratio of the areas of the triangles A, O, B and C, O, D. Before moving on to the solution, let us first understand the key idea which would help us to solve this problem. Now the key idea is triangle say PQR is similar to triangle say DEF. The ratio of the areas of the similar triangles that is area of triangle PQR upon area of triangle DEF is equal to the square of the ratio of their corresponding sides. That is area of triangle PQR upon area of triangle DEF is equal to PQ square upon DE square is equal to QR square upon EF square is equal to this becomes the key idea. Let's now proceed on with the solution. Now we have to find the ratio of the areas of the triangle A, O, B and C, O, D. Where we are given that A, B, C, D is a trapezium in which A, B is parallel to C, D and A, B is twice of C, D. So we are given B is twice of C, D. Let us name this as one. Now we have to consider the triangles A, O, B and C, O, D. So in triangle A, O, B, C, O, D, A, O, B is equal to angle C, O, D to this angle because these are vertically opposite angles. Again we are given that A, B is parallel to, so this implies angle A, B that is this angle is equal to angle O, C, D that is this angle O, C, D. Since these are alternate angles triangles are equal therefore by angle angle criteria is A, A criteria of similarity triangle A, O, B is similar to triangle C, O, D. Now we will be using the theorem. So triangle A, O, B is similar to triangle C, O, D. Therefore ratio of the area of the two similar triangles that is area of triangle A, O, B upon area of triangle C, O, D is equal to the square of the ratio of the corresponding side. That is A, B square upon DC square. Now we know that A, B is twice of CT. So from one we will be having area of triangle A, O, B upon area of triangle C, O, D is equal to twice of DC square upon DC square. By one. So from this we have area of triangle A, O, B upon area of triangle C, O, D is equal to 4 upon 1 hence the required ratio is 4 is to 1. That is area of triangle A, O, B is to area of triangle C, O, D given by 4 is to 1. So this completes the question and the session. Bye for now. Take care. Have a good day.