 Hello and welcome to the session. Let's work out the following problem. It says prove that the area of an Equilateral triangle Described on one side of a square is equal to half the area of the Equilateral triangle described on one of its diagonal. So let's now move on to the solution let A, B, C, D be a square P, A, D and Q, A, C are two equilateral triangles described on side A, D and the diagonal A, C respectively. Now we have to prove that the area of triangle P, A, D is half the area of the triangle Q, A, C as Triangle Q, A, C is described on the diagonal A, C and triangle P, A, D is described on this side A, D of the square, A, B, C, D and let side of the square be So let A, B, C, D be a square T, A, D is the triangle in fact is an it's an equilateral triangle described on the side A, D and Q, A, C is the equilateral triangle described on the diagonal A, C. Now A, D, C is the right angles triangle in which A, C is the hypotenuse. So A, C square is equal to A, D square plus D, C square that is A square plus A square that is to A square. This is by Pythagoras theorem. Now Triangle A, D is similar to triangle Q, A, C since both triangles are equilateral triangle and we know that in equilateral triangle each angle is 60 degrees. So these two triangles are similar by angle angle angle similarity. Each angle is 60 degrees. Now we know that ratio of areas of two similar triangles is equal to the square of ratio of their corresponding sides. So we have area of triangle P, A, D upon area of triangle Q, A, C is equal to the square of ratio of corresponding sides that is A, D square upon A, C square right. So this implies area of triangle P, A, D upon area of triangle Q, A, C is equal to A, D square. Now A, D is the side of the square. So this is A square and A, C square is 2A square. So this implies area of triangle P, A, D is equal to 1 by 2 into area of triangle Q, A, C that is area of triangle P, A, D is half the area of triangle Q, A, C. Hence the result is proved. So this completes the question and the session. Bye for now. Take care and have a good day.