 Hi and welcome to the session. Let us discuss the following question. Question says, you have studied that a median of a triangle divides it into two triangles of equal areas. Verify this result for triangle ABC whose vertices are A, 4 minus 6, B, 3 minus 2 and C, 5, 2. Let us now start with the solution. We have given a triangle ABC where coordinates of A are 4 minus 6, coordinates of B are 3 minus 2 and coordinates of C are 5, 2. Now let us draw median AD on BC. We know median of a triangle divides the opposite side into two equal parts. So this implies B is the midpoint of BC. Now we can write that we are given a triangle ABC whose vertices are A, 4 minus 6, B, 3 minus 2 and C, 5, 2. Now we know AD is the median on BC. This implies BD is equal to DC. We know median divides the opposite side into two equal parts. So therefore, D is the midpoint of BC. Now first of all let us find out coordinates of D. We know that coordinates of D are 3 plus 5 upon 2 and minus 2 plus 2 upon 2. We can say it is equal to 4, 0. To find the coordinates of D we have used the midpoint formula that is X1 plus X2 upon 2, Y1 plus Y2 upon 2. Midpoint of a line joining points X1, Y1 and X2, Y2 is given by this formula. Now we get coordinates of D are 4, 0. Now we have to show that area of triangle ABD is equal to area of triangle ACD. We know area of triangle pound by vertices X1, Y1, X2, Y2 and X3, Y3 is given by 1 upon 2 multiplied by X1, Y2 minus Y3 plus X2 multiplied by Y3 minus Y1 plus X3 multiplied by Y1 minus Y2. Now first of all let us consider triangle ABD. Let A represents coordinates X1, Y1, B represents coordinates X2, Y2 and D represents coordinates X3, Y3. Now clearly we can see X1 is equal to 4, Y1 is equal to minus 6, X2 is equal to 3, Y2 is equal to minus 2, X3 is equal to 4, Y3 is equal to 0. So we can write area of triangle ABD is equal to 1 upon 2 multiplied by X1, X1 we know is equal to multiplied by Y2 minus Y3, Y2 is equal to minus 2 and Y3 is equal to 0. So minus 2 minus 0 plus X2 multiplied by we know X2 is equal to 3 multiplied by Y3, we know Y3 is equal to 0 minus Y1, Y1 is equal to minus 6. So we can write here minus minus 6 plus X3, we know X3 is equal to 4 multiplied by Y1 minus Y2, Y1 is equal to minus 6 minus minus 2, we know Y2 is equal to minus 2. Now simplifying further we get 1 upon 2 multiplied by 4 multiplied by minus 2 plus 3 multiplied by 6 plus 4 multiplied by minus 6 plus 2 we know minus 2 minus 0 is equal to minus 2 negative sign will change into positive sign and we get 0 plus 6 is equal to 6, here 2 negative signs will change into positive sign, here we get 1 upon 2 multiplied by minus 8 plus 18 plus 4 multiplied by minus 4 we know minus 6 plus 2 is equal to minus 4. Now this is further equal to 1 upon 2 multiplied by we know minus 8 plus 18 is equal to 10, 10 and 4 multiplied by minus 4 is equal to minus 16 and 10 minus 16 is equal to minus 6. So we get 1 upon 2 multiplied by minus 6, now this is further equal to minus 3. So we get area of triangle ABD is equal to minus 3, we know area can never be negative since area cannot be negative we will take the numerical value of minus 3 that is 3. So we get area of triangle ABD is equal to 3 square units, now we will find out area of triangle ADC that A represents the point X1 Y1, D represents the point X2 Y2 and C represents X3 Y3. Now here X1 is equal to 4, Y1 is equal to minus 6, X2 is equal to 4, Y2 is equal to 0, X3 is equal to 5 and Y3 is equal to 2. Now area of triangle ADC is equal to 1 upon 2 multiplied by X1 we know X1 is equal to 4 multiplied by Y2 minus Y3, Y2 is equal to 0 minus 2, you know Y3 is equal to 2 plus X2, X2 we know is equal to 4. So we will write 4 here multiplied by Y3 minus Y1, Y3 is equal to 2 and Y1 is equal to minus 6. So we can write 2 minus minus 6 plus X3, X3 we know is equal to 5 multiplied by Y1 minus Y2, we know Y1 is equal to minus 6 and Y2 is equal to 0. So we can write minus 6 minus 0 here. Now this is further equal to 1 upon 2 multiplied by 4 multiplied by minus 2 we know 0 minus 2 is equal to minus 2 plus 4 multiplied by 2 plus 6 here you know minus minus sign will change into positive sign plus 5 multiplied by minus 6 minus 6 minus 0 is equal to minus 6. Now we get 1 upon 2 multiplied by minus 8 plus we know 6 plus 2 is equal to 8 and 8 multiplied by 4 is equal to 32 minus 30 minus 6 multiplied by 5 is equal to minus 30. Now simplifying further we get 1 upon 2 multiplied by minus 6, now this is equal to minus 3. Now we know area cannot be negative so we will take the numerical value of minus 3 that is 3. So we get area of triangle ADC is equal to 3 square units. Now clearly we can see area of triangle ABD is equal to area of triangle ADC is equal to 3 square units. So this shows that a median of a triangle divides it into two triangles of equal areas. So hence verified this completes the session hope you understood the session take care and have a nice day.