15.Eis the mid-point of the non-parallel side BC of a trapezium ABCD.E...
To prove that the area of triangle ABE is equal to half the area of trapezium ABCD, we can use the formula for the area of a trapezium:
Area of trapezium ABCD = (sum of lengths of parallel sides) x (height) / 2
Since E is the midpoint of side BC, we can call the length of side BC "2x" and the length of side AD "2y". We can also call the distance between points A and D "h".
Substituting these values into the formula for the area of a trapezium, we get:
Area of trapezium ABCD = (2x + 2y) x h / 2
Now, let's consider the area of triangle ABE. Since E is the midpoint of side BC, we know that the length of side AE is equal to x. We also know that the length of side AB is equal to y and the height of triangle ABE is equal to h.
Substituting these values into the formula for the area of a triangle (base x height / 2), we get:
Area of triangle ABE = xh / 2
Now, we can set the expression for the area of triangle ABE equal to half the expression for the area of trapezium ABCD and solve for x:
xh / 2 = (2x + 2y) x h / 4
x = y
This means that if the lengths of sides AB and AD are equal, then the area of triangle ABE is equal to half the area of trapezium ABCD.
The same logic can be applied to triangle DCE to prove that the area of triangle DCE is also equal to half the area of trapezium ABCD. Therefore, the overall result is:
Area of triangle ABE + Area of triangle DCE = 2 x (half the area of trapezium ABCD)
Which simplifies to:
Area of triangle ABE + Area of triangle DCE = Area of trapezium ABCD
Thus, we have proven that the area of triangles ABE and DCE is equal to the area of trapezium ABCD.