 Hello and welcome to the session. In this session we discuss the following question it says, D is the midpoint of side BC of triangle ABC and E is the midpoint of BD. If O is the midpoint of AE, prove that area of triangle BOE is equal to 1 upon 8 area of triangle ABC. This is the figure given to us. Before we move on to the solution let's recall a fact which says that a median of a triangle divides it into two triangles of equal areas. This is the key idea for this question. Now we move on to the solution. This is the figure given to us. In this we have that D is the midpoint of the side BC then E is the midpoint of the side BD then O is the midpoint of the side AE and we need to prove that the area of triangle BOE is equal to 1 upon 8 into area of the triangle ABC. Now since we have that B is the midpoint of the side BC so this means that AD is the median and therefore it would divide the triangle ABC into two triangles of equal area. So we say that area of the triangle ABD is equal to the area of the triangle ADC. Now the area of the triangle ABC is equal to area of the triangle ABD plus the area of triangle ADC. Area of triangle ABD and area of triangle ADC are equal. So from here we get area of triangle ABC is equal to two times the area of triangle ABD that is area of the triangle ABD is equal to half the area of triangle ABC. Let this be equation one. Now again since we know that E is the midpoint of the side BD so this means that AE is the median of triangle ABD. Now since we know that the median of a triangle divides the triangle into two triangles of equal area therefore we get area of triangle ABE is equal to the area of the triangle AED that is we have area of the triangle ABE is equal to half the area of triangle ABD let this be equation two. Now using equation one in equation two we get area of triangle ABE is equal to half the area of triangle ABD which is half the area of triangle ABC. So half into half of area of triangle ABC so this is equal to one fourth the area of triangle ABC that is area of triangle ABE is equal to one fourth the area of triangle ABC. We take this as equation three. Now next we have that O is the midpoint of the side AE therefore we have that BO is the median of triangle ABE and we know that median of a triangle divides the triangle into two triangles of equal area so this means that area of the triangle BOE is equal to the area of the triangle AOB or we can say that area of the triangle BOE is equal to half the area of triangle ABE. We take this as equation four now using equation three in equation four we get that area of triangle BOE is equal to half the area of triangle ABE which is one fourth the area of triangle ABC. So we get area of triangle BOE is equal to one upon eight into area of triangle ABC and we were supposed to prove this only so hence proved this complete BC session hope you understood the solution for this question.