In an isosceles trapezoid with bases 40 and 24, the diagonals are perpendicular. find the area.

Let’s add a straight line MK, which passes through the midpoints of the bases. For an isosceles trapezoid, it will be the axis of symmetry.

MK will divide the ВOС and AOD triangles into equal triangles.

Let us prove that ΔВМО and ΔМСО are isosceles.

<MBO = <MCO = (180 ° – 90 °) / 2 = 45 °.

The MK axis of symmetry divides the BOC angle into two equal angles:

<BOM = MOC = 90 ° / 2 = 45 °.

The angles at the bases BO and OC are equal, which means that the corresponding triangles are isosceles.

OM = MC = BC / 2 = 24/2 = 12.

Similarly, one can prove that OK = KD = 40/2 = 20.

Height of MK trapezoid:

MK = OM + OK = 12 + 20 = 32.

Half sum of bases:

(AD + BC) / 2 = (40 + 24) / 2 = 32.

Trapezium area:

S = 32 * 32 = 1024.

Answer: 1024.