In trapezoid ABCD, the length of the bases is AD = 24 cm, BC = 8 cm, the length

In trapezoid ABCD, the length of the bases is AD = 24 cm, BC = 8 cm, the length of the diagonals AC = 3 cm, BD = 5cm. Find the area of the trapezoid.

In the task, using the known lengths of the bases and the lengths of the diagonals of the trapezoid, determine its area. This kind of problem is solved as follows. Let a trapezoid ABCD be given, where AC || BD, AD and BC are its diagonals. Let’s draw a parallel line to any diagonal (for example, to the diagonal BD), passing through any other vertex (for example, through the vertex C). Then a triangle equal in size with the trapezoid is formed, the base of which will be equal to the sum of the bases of the trapezoid, and the heights of the trapezoid and the triangle will be equal to each other. The area of ​​a triangle is easily determined (for example, using Heron’s formula), which will be equal to the required area of ​​the trapezoid.
However, there are certain rules by which you can build a certain geometric figure. For example, for any triangle: the sum of any two of its sides is strictly greater than the third side. When it comes to a trapezoid, then there are also conditions for existence for it. For example, for any trapezoid ABCD, where AC || BD, AD and BC are its diagonals, the equality must be true: AC ^ 2 + BD ^ 2 – 2 * AD * BC = AB ^ 2 + CD ^ 2.
Let us check this equality for the data of the task AD = 24 cm, BC = 8 cm, AC = 3 cm, BD = 5 cm. Let us calculate the left side of the equality. We have (for brevity, omit the unit of measurement cm) 3 ^ 2 + 5 ^ 2 – 2 * 24 * 8 = 9 + 25 – 384 = –350 <0. However, the right side of the equality as the sum of two positive numbers AB2 + CD2> 0 This contradiction shows that there is no trapezoid for which AD = 24 cm, BC = 8 cm, AC = 3 cm, BD = 5 cm.
Answer: The terms of the assignment are contradictory.