 Hello friends, welcome to the session. I am Malka. Let's discuss the given question. Let A with the coordinates 4 to B coordinates 6, 5 and C with the coordinates 1, 4 be the vertices of triangle A, B, C. First path is the medium from A meets B, C at D. Find the coordinates of the point D. Our figure is A, B, C is the triangle and A, D is the medium on B, C. Now let's discuss the first path. Since we are given that the medium from A meets B, C at D. So we have to find the coordinates of the point D and D is the midpoint of B, C. Therefore, we will apply the midpoint formula. That is midpoint formula is given by x1 plus x2 upon 2 and y1 plus y2 upon 2. So this will give the coordinates of the midpoint. That is the point D. So coordinates of DR 6 plus 1 by 2 and 5 plus 4 by 2. So coordinates of DR 7 by 2 and 9 by 2. So this is the solution of the first path. Now let's see the next path. Our next path is find the coordinates of point P on A, D. There is a point P on A, D such that A, P is to PD equal to 2 is to 1. Now let's begin with the solution of the path second. P is the point on A, D such that A, P is to PD equal to 2 is to 1. Now we have to find the coordinates of the point P and we are given the coordinates of the point A and D. So we will use the section formula to find the coordinates of the point P and we know that section formula m1 x2 plus m2 x1 upon m1 plus m2 m1 by 2 plus m2 y1 upon m1 plus m2. So these are the coordinates of any point which divide the line segment in the ratio of m1 is to m2. Therefore coordinates of P will be into 7 by 2 plus 1 into 4 upon 2 plus 1 into 9 by 2 plus 1 into 2 upon 2 plus 1. So these are the coordinates of the point P. On further simplifying them we get 11 upon 3 and 11 upon 3 as the coordinates of the point P. So hope you understood the solution of the second path. Our third path is find the coordinates of points Q and R on mediums VE and CF respectively such that VE VQ is to QE equal to 2 is to 1 and CR is to RF equal to 2 is to 1. Now let's begin with the solution of the path third. Here we have to find the coordinates of point E and F. So coordinates of will be, here we will use the midpoint formula that is x1 plus x2 upon 2. So it will be 4 plus 1 upon 2 and y1 plus y2 upon 2 that is 2 plus 4 upon 2. So this will give us 5, 5 by 2 and 6 by 2 or we can say 5 by 2 and 3. These are the coordinates of the point E. Now we find the coordinates of point Q and we have given that VQ is to QE equal to 2 is to 1. So coordinates Q will be, now here again we will use the action formula. So that is 2 into 5 by 2 plus 1 into 6 upon 2 plus 1 and the y coordinate will be 2 into 3 plus 1 into 5 upon 2 plus 1. So these are the coordinates of the point Q which on further simplification will be 11 upon 3 and 11 upon 3. So these are the coordinates of the point Q. Now we have to find the coordinates of the point R. So for this we first find the coordinates of the point F. Therefore coordinates of will be, it will be given by the midpoint formula. So 6 plus 4 by 2 and 5 plus 2 by 2 will give us the coordinates of the point F or it can be written as 5 and 7 by 2. These are the coordinates of the point F. Now we find the coordinates of the point R and we have also given that C R is to RF equal to 2 is to 1. Therefore coordinates of will be 2 into 5 plus 1 into 1 upon 2 plus 1 into 7 by 2 plus 1 into 4 upon 2 plus 1. So these are the coordinates of the point R. Which on further simplification will be 11 upon 3 and 11 upon 3. So these are the coordinates of the point R. Hope you understood the solution of the third part. Now let's discuss the fourth part. Our fourth part is what do you observe? As we have seen that the coordinates of the point P, Q and R are same. So this implies that P, Q and R is same point. So we observe that coordinates of P, Q same. So therefore this implies that Q point, the centroid divides the median in the ratio 2 is to 1. Centroid divides the median in the ratio 2 is to 1. So this is the observation we had from our figure. Now let's discuss the next part. Our fifth part is F A with the coordinates x1, y1, b, coordinates x2, y2 and c with coordinates y3. Our vertices of triangle A, B, C find the coordinates of the centroid of the triangle. Let's begin with the solution of the part fifth. A, B, C is the triangle with coordinates of A as x1, y1, b, x2, y2, c, x3, y3, ad, be and cf are the medians and the common point of intersection of the medians is the point P which is the centroid and we have to find the coordinates of centroid. So to find the coordinates of the point P, we first have to find the coordinates of the point D and we know that the centroid divides the median in the ratio 2 is to 1. So let's find the coordinates of the point D, coordinates, we find the coordinates of D by the midpoint formula. So that is, we can see from the figure x2 plus x3 by 2 and y2 plus y3 by 2. So these are the coordinates of the point D. Now we have to find the coordinates of the point P and we know that if P is the centroid then the ratio of AP is to PD equal to 2 is to 1. So we can write AP is to PD equal to 2 is to 1. Now we will find the coordinates of, therefore the coordinates of P are, we will find the coordinates of P by the section formula that is 1 into x1 plus 2 into x2 plus x3 by 2 upon 1 plus 2 and the y coordinate is given by 1 into y1 plus 2 into y2 plus y3 by 2 upon 1 plus 2. So on for the simplification it can be written as x1 plus x2 plus x3 by 3 and y1 plus y2 plus y3 by 3. So these are the coordinates of the point P that is the centroid. Therefore coordinates of centroid P x1 plus x2 plus x3 by 3 and y1 plus y2 plus y3 by 3. So hope you understood this part. Answer to the different parts of the question are D coordinates of D are 7 by 2 and 9 by 2, coordinates of the point P are 11 by 3, 11 by 3, coordinates of the point Q and R are 11 by 3, 11 by 3 and the basic observation is that P, Q and R are the same points and the coordinates of the centroid at this point P are given by x1 plus x2 plus x3 by 3, y1 plus y2 plus y3 by 3. So these are the coordinates of the centroid. So hope you understood the solution of all the parts and enjoyed the session. Goodbye and take care.