Find the side of an isosceles trapezoid whose bases are 12 cm and 6 cm, and one of the corners is 45 degrees.

A trapezoid is a quadrilateral in which one pair of opposite sides is parallel, and the sides are not equal to each other.
Isosceles is a trapezoid in which the sides are equal:
AB = СD.
In order to calculate the length of the side side of the trapezoid, consider the triangle ΔАВН.
Since the segment of the larger base, located between the heights of the trapezoid, is equal to the length of its smaller base, then:
НK = BC;
AH = KD = (AD – BC) / 2;
AH = KD = (12 – 6) / 2 = 6/2 = 3 cm.
To calculate AB, we apply the cosine theorem. The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
cos A = AH / AB;
AB = AH / cos A;
cos 45º ≈ 0.707;
AB = 3 / 0.707 ≈ 4.2 cm.
Answer: the length of the sides AB and СD is 4.2 cm.