One of the bases of the trapezium is 2 times larger than the other, the angle at the base is 90 * and 45 *

One of the bases of the trapezium is 2 times larger than the other, the angle at the base is 90 * and 45 *. Find the lateral sides of the trapezoid if the base is less than 12 cm.

By condition, the smaller base a is 12 cm, which means the larger base b is 12 * 2 = 24 cm.

The angles for a larger base are 90 ° and 45 °, therefore, this trapezoid is rectangular, which means its height h coincides with the smaller side side.

The larger base is equal to the sum of the lengths of the smaller base and the projection of the inclined lateral side c onto the larger base. Hence, the projection of the inclined side is equal to the difference between the lengths of the larger and smaller bases: 24 – 12 = 12 cm.

Consider a right-angled triangle formed by the height of the trapezoid h, the inclined side c and its projection. The angle between the oblique side and the large base is 45 °. The height is the leg opposite to this angle, the projection equal to 12 cm is adjacent. The ratio of the opposite leg to the adjacent leg is the tangent of the angle:

tg 45 ° = h / 12;

h = 12 * tg 45 ° = 12 * 1 = 12 cm – the smaller side of the trapezoid.

The ratio of the adjacent leg to the hypotenuse is the cosine of the angle:

cos 45 ° = 12 / s;

c = 12 / cos 45 ° = 12 / (√2 / 2) = 12√2 cm – large side.