 Hello friends, welcome to the session. I am Anka. Let's discuss the given question. Are practice if A, B and P, M are millions of a triangle A, B, C and P, Q, R respectively where triangle A, B, C is similar to triangle P, Q, R. Prove that A, B upon P, Q equal to A, B upon P, M. Now let's start with the solution. Here are the figures according to the question where A, B, C and P, Q, R are the triangles and A, B and P, M are millions of the triangle A, B, C and P, Q, R respectively. So we are given that triangle A, B, C is similar to triangle P, Q, R and A, D is the medium of triangle A, B, C and P, M is the medium of triangle P, Q. Now we have to prove that A, B upon P, Q equal to A, D upon P, M. Now let's start with the proof. A, D is the medium of the triangle A, B, C. So it will divide the opposing side that is B, C into equal parts and similarly P, M divides the opposing side Q, R in equal parts. This implies B, B equal to half of V, C, Q, M equal to half of Q, R. So let this be our first equation. Now we are also given that triangle A, B, C is similar to triangle P, Q, R. This is given to us. Therefore this implies that angle B is equal to angle Q. Let this be our second equation and since the two triangles are similar so this implies A, B upon P, Q equal to B, C upon Q, R equal to A, C upon P, R. This implies A, B upon P, Q equal to, now we will substitute the value of B, C from our first equation that is V, C equal to 2, V, D and Q, R equal to 2, Q, M equal to A, C upon P, R. Here we see that 2, 2 cancel out. This can be written as A, B upon P, Q equal to B, D upon Q, M. Let this be our third equation. Now in triangle A, B, D and P, Q, M we have angle B equal to angle Q from our equation second, angle B equal to angle Q from equation second and we also know that A, B upon P, Q equal to V, D upon Q, M from third equation. So therefore we can say that by SAS criteria of similarity triangle A, B, D is similar to triangle P, Q, M. Therefore by SAS criteria the triangle ABD is similar to triangle P, Q, M. Since we all know that in case of two similar triangles the ratio of the corresponding sides are equal. So this implies AB upon P, Q equal to BD upon Q, M equal to AD upon P, M. This implies AB upon P, Q equal to AD upon P, M as proved. Hope you got the solution and enjoyed the session. Goodbye and take care.