The bases of the rectangular trapezoid are 26 cm and 36 cm. The large diagonal of the trapezoid is the bisector

The bases of the rectangular trapezoid are 26 cm and 36 cm. The large diagonal of the trapezoid is the bisector of an acute angle. Find the perimeter of the trapezoid.

At the intersection of parallel intersecting lines in a cross, the lying angles are equal. Since the diagonal is a secant for the parallel bases of the trapezoid, and is also the bisector of an acute angle, the angles between the larger diagonal and the larger base, between the larger diagonal and the smaller base, between the larger diagonal and the lateral inclined side are equal. Thus, in a triangle formed by a diagonal, a smaller base and a side sloped side, the two angles are equal. Therefore, this triangle is isosceles with a base equal to the larger diagonal of the trapezoid. The smaller base of the trapezium and its lateral side are equal to each other, like the lateral sides of an isosceles triangle. This means that the lateral inclined side of the trapezoid is equal to its smaller base and is equal to 26 cm.

The length of the larger base of the trapezoid is equal to the sum of the lengths of its smaller base and the projection of the lateral inclined side onto the larger base. Hence, the projection of the lateral side is equal to the difference in the lengths of the bases: 36 – 26 = 10 cm.

From a right-angled triangle formed by the lateral side of the trapezoid, its height and the projection of the lateral side onto a larger base, according to the Pythagorean theorem, we can find the height of the trapezoid: =

h ^ 2 = 26 ^ 2 – 10 ^ 2 = 676 – 100 = 576;

h = √576 = 24 cm.

In a rectangular trapezoid, its smaller lateral side, perpendicular to the bases, is equal to the height of the trapezoid.

Thus, we can find the perimeter of the trapezoid: P = 26 + 36 + 26 + 24 = 112 cm.