Find the diagonals of an isosceles trapezoid if the smaller base is 7, the lateral base is 5√2

Find the diagonals of an isosceles trapezoid if the smaller base is 7, the lateral base is 5√2, and one of the angles of the trapezoid is 135 degrees.

Let’s build the heights BH and CH of an isosceles trapezoid.

In a right-angled triangle ABH, the angle ABH = ABC – CBH = 135 – 90 = 45, then triangle ABH is isosceles, AH = BH.

AB ^ 2 = 2 * AH ^ 2 = 2 * BH ^ 2.

2 * AH ^ 2 = 50.

AH ^ 2 = 25.

AH = BH = 5 cm.

Rectangular triangles ABH and CDM are equal in hypotenuse and acute angle, then DM = AH = 5 cm.

Quadrangle BCMН is a rectangle, then НM = BC = 7 cm, and DH = DM + НM = 5 + 7 = 12 cm.

In a right-angled triangle BDH, according to the Pythagorean theorem, BD ^ 2 = BH ^ 2 + DН ^ 2 = 25 + 144 = 169.

BD = 13 cm.

Answer: The length of the diagonal is 13 cm.