An isosceles trapezoid is given. The bases are 4 and 12 cm. The side is 5 cm. Find the height and diagonal.

1) Draw the trapezoid to a rectangle, we get points A ‘and D’. The length of A’D ‘will be equal to the base of AD, that is, 12 cm.

2) Since the trapezoid is equilateral, then A’B = CD ‘= (12 – 4): 2 = 4 cm.

3) The resulting triangle AA’B is Egyptian, since it is rectangular and its sides are proportional to 3: 4: 5. By the Pythagorean theorem, we find that the height of the trapezoid will be 3 cm.

4) In order to find the diagonal of a trapezoid, it is necessary to use the basic property of the diagonals of a trapezoid, which sounds like this: the sum of the squares of the diagonals of a trapezoid is equal to the sum of the squares of the sides plus twice the product of its bases.

AB ^ 2 + CD ^ 2 = AB ^ 2 + CD ^ 2 + 2 * AD * BC; since AB = CD, then

2 * AB ^ 2 = 5 ^ 2 + 5 ^ 2 + 2 * 12 * 4 = 25 + 25 + 96 = 146

AB ^ 2 = 146: 2

AB = 73 ^ (1/2) = 8.544

Answer: the height of the trapezoid is 3 cm, the diagonal is 8.544 cm.