 Hi and welcome to the session. Let's discuss the following question. It says 10 right triangle A, B, C. Right angle at C. M is the midpoint of the hypotenuse A, B. C is joined to M and produced to a point D such that B, M is equal to C, M. Point D is joined to B. Show that triangle A, M, C is congruent to triangle B, M, D. Angle D, B, C is a right angle. Triangle D, B, C is congruent to triangle A, C, B and C, M is half of A, B. To solve this problem we will be using SAS congruence criteria, which is the T idea. Now we are given in triangle A, B, C angle A, C, B is 90 degree and M is the midpoint of A, B is equal to B, M. And we are also given that D, M is equal to C, M. Let's now move on to the solution. In the first part we have to show triangle A, M, C is congruent to triangle B, M, D. Now is equal to B, M because M is the midpoint of A, B and angle B, M, D is equal to angle A, M, C because they are vertically opposite angles. We are given that C, M is equal to D, M. Hence by SAS congruence criteria, triangle A, M, C is congruent to triangle B, M, D. Let's now move on to the second part. We have to prove that angle D, B, C is a right angle. Now as triangle A, M, C is congruent to triangle B, M, D. Therefore, angle C, A, B is equal to angle D, B, A right. Because as we know that whenever two triangles are congruent, their corresponding parts are congruent. So by C, P, C, T, C, two angles are equal but they are alternate angles. Therefore B, D is parallel to A, C right and which implies that sum of angle B and C is 180. This is because sum of interior angles on the same side of transversal 180. Their transversal is BC. So this implies angle B is equal to 180 minus angle C. Now angle C is given to be 90 degrees. So angle B is equal to 180 minus 90 that is 90 degrees. So this implies angle D, B, C is a right angle. Let's now discuss the third part. We have to prove that triangle D, B, C is congruent to triangle AC, B. Now D, B is equal to AC as triangle A and C and B and D are congruent. So as triangle A, M, C is congruent to triangle B and D. Therefore D, B is equal to AC angle. D, B, C is equal to angle AC, B is equal to 90 degrees. This we have proved above. Also BC is equal to BC since it's the common side. So we have proved that two sides and the included angle of the two triangles are equal. Therefore by SAS criteria triangle D, B, C is congruent to triangle AC, D by SAS congruence criteria. Hence we have proved that triangle D, B, C is congruent to triangle AC, B. So this completes the third part. Let's now discuss the fourth part in which we have to show that C, M is half of AB. Now as triangle D, B, C is congruent to triangle AC, B. Therefore DC is equal to AB by CP, CT. That is corresponding curves of congruent triangles are congruent. Also D, M is equal to C, M it is given to us. And also AM is equal to MB because M is the midpoint of AB. Therefore DC and AB by six each other. Hence AM is equal to 1 by 2 of AB. So this completes the question. But for now take care, hope you enjoy the session.