 Hello and welcome to the session. Let us discuss the following question. Question says, find the area of the triangle formed by joining the midpoints of the sides of the triangle whose vertices are 0, minus 1, 2, 1 and 0, 3. We have to find the ratio of this area to the area of the given triangle. Now, let us start the solution. First of all, let us consider a triangle ABC whose vertices are 0, minus 1, B, 2, 1 and C, 0, 3. This is the triangle ABC. A has coordinates 0, minus 1, B has coordinates 2, 1 and C has coordinates 0, 3. Now, let us assume that DE and F are midpoints of AB, BC and CA. Now, by joining DE, EF and DF, we get the triangle DF. Now, to find the area of triangle DEF, first of all, we will find out coordinates of D, coordinates of E and coordinates of F. Now, first of all, let us find out coordinates of D. We know D is the midpoint of AB and we know if we are given a line PQ such that coordinates of PR, X1, Y1 and coordinates of QR, X2, Y2, then PQ has midpoint R whose coordinates are X1 plus X2 upon 2, Y1 plus Y2 upon 2. So, we will use this formula to find out the coordinates of D. Now, using the midpoint formula, we get coordinates of DR, 0 plus 2 upon 2, minus 1 plus 1 upon 2. So, we can write coordinates of DR, 0 plus 2 upon 2, minus 1 plus 1 upon 2. Or we can say, coordinates of DR, 1, 0. Similarly, we can find coordinates of E. We know E is the midpoint of BC. So, coordinates of BR, 2 plus 0 upon 2, 1 plus 3 upon 2. Or we can say, coordinates of BR, 1, 2. Now, we know F is the midpoint of AC. So, we will find coordinates of F now. So, 0 plus 0 upon 2 and 3 plus minus 1 upon 2 are coordinates of F. Or we can simply write it as 0. We know 3 minus 1 is equal to 2. So, 2 upon 2 is equal to 1. So, we get coordinates of FR, 0, 1. Now, we have to find the area of triangle, DEF. Now, first of all, let us recall that area of a triangle having vertices x1, y1, x2, y2 and x3, y3 is given by half multiplied by x1 multiplied by y2 minus y3 plus x2 multiplied by y3 minus y1 plus x3 multiplied by y1 minus y2. Now, we will use this formula to find the area of the triangle, DEF. Let us assume that coordinates of BR, x1, y1, coordinates of ER, x2, y2 and coordinates of FR, x3, y3. Now, clearly we can see x1 is equal to 1, y1 is equal to 0, x2 is equal to 1, y2 is equal to 2, x3 is equal to 0 and y3 is equal to 1. Now, we will substitute all these corresponding values in this formula. Now, we can write area of triangle, DEF is equal to half multiplied by x1. We know is equal to 1 multiplied by y2 minus y3, y2 is equal to 2 and y3 is equal to 1. So, we will write 2 minus 1 here plus x2. We know x2 is equal to 1 multiplied by y3 minus y1 that is 1 minus 0 plus x3. x3 we know is equal to 0 multiplied by y1 minus y2, y1 is equal to 0 and y2 is equal to 2. So, here we will write 0 minus 2. Now, simplifying we get 1 upon 2 multiplied by 1. We know 2 minus 1 is equal to 1 and 1 multiplied by 1 is 1 plus 1 minus 0 is 1 and 1 multiplied by 1 is equal to 1. So, we will write 1 here. Now, 0 multiplied by any number is equal to 0. So, we will write 0 here. Now, we get 1 upon 2 multiplied by 2 which is further equal to 1. So, we get area of triangle DEF is equal to 1 square units. This completes the first part of the question. Now, we have to find ratio of the area of triangle DEF to triangle ABC. Now, we note area of triangle DEF is equal to 1 square units. Now, we will find out area of triangle ABC. Now, again we will use this formula to find the area of triangle ABC. Now, let us assume that coordinates of AR, X1, Y1, coordinates of BR, X2, Y2 and coordinates of CER, X3, Y3. Now, we will write area of triangle ABC is equal to half multiplied by X1. X1 we know is equal to 0 multiplied by Y2 minus Y3. Y2 is equal to 1 and Y3 is equal to 3. So, we will write here 1 minus 3 plus X2 we know X2 is equal to 2. So, we will write here 2 multiplied by Y3 minus Y1. We know Y3 is equal to 3 and Y1 is equal to minus 1. So, we will write here minus minus 1 plus X3. X3 is equal to 0 multiplied by Y1 minus Y2. Y1 is minus 1 and Y2 is 1. So, we will write here minus 1 minus 1. Now, simplifying we get area of triangle ABC is equal to half multiplied by 0 plus 2 multiplied by 3 plus 1 plus 0. We know 0 multiplied by any number is equal to 0. Here also 0 multiplied by any number is equal to 0. Here we know minus minus sign become positive sign. Now, this is further equal to half multiplied by 8. We know 3 plus 1 is equal to 4 and 4 multiplied by 2 is equal to 8. So, simplifying further we get area of triangle ABC is equal to 4 square units. Now, let us find out required ratio that is area of triangle DEF upon area of triangle ABC. We know area of triangle DEF is equal to 1 square units and area of triangle ABC is equal to 4 square units. So, the required ratio is 1 is to 4. So, this is our required answer. This completes the session. Hope you understood the session. Take care and keep smiling.