 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, A B C D is a trapezium in which A B is parallel to D C and its diagonals intersect each other at point O. Show that A upon B O is equal to C O upon D O. First of all let us understand basic proportionality theorem. Basic proportionality theorem states that for line is drawn parallel to one side of a triangle intersecting other two sides at two distinct points then the other two sides are divided in the same ratio. That is if A B C is a triangle given to us and D E is parallel to B C and D E is intersecting A B and A C at two distinct points that is D and D then A D upon D B is equal to A E upon E C. This is the key idea to solve the given question. Now let us start the solution. First of all let us write what all is given to us in the question. This is a trapezium in which A B is parallel to C D. Now we have to prove that A O upon D O is equal to C O upon D O. Now before starting the proof we will do some construction. We will draw O E parallel to A B. Now here we have drawn O E parallel to A B. So we can write construction through O draw O E parallel to A B. Let us start the proof now. Now we know O E is parallel to A B by construction and also A B is parallel to C D. This is given in the question. So this implies O E is parallel to C D. O E is parallel to A B as well as C D is parallel to A B. Now two lines parallel to same given line are parallel to each other. Now let us consider triangle A D C. Triangle A D C. O E is parallel to C D. This we have shown above. My basic proportionality theorem we get. A E upon A B is equal to A O upon O C. Now let us consider triangle A B D. In triangle A B D O E is parallel to A B. My basic proportionality theorem we get. D E upon E A is equal to D O upon O B. Taking reciprocal of both the ratios we get A E upon E D is equal to O B upon O D. In triangle A D C A E upon E D is equal to A O upon O C. Let us name this expression as 1 and in triangle A B D A E upon E D is equal to O B upon O D. Let us name this expression as 2. Now from 1 and 2 we get A O upon O C is equal to O B upon O D. Since both the ratios are equal to A E upon E D. Now we can write A O upon O C is equal to O B upon O D. Multiplying both sides by O C upon O B we get. A O upon O B is equal to O C upon O D. We can rewrite this as A O upon B O is equal to C O upon D O. So A O upon B O is equal to C O upon D O. Is there a required answer? This completes the session. Hope you understood the session. Take care and have a nice day.