Question

Углы при одном из оснований трапеции, равны 42° и 48°, а отрезки, соединяющие середины противоположных сторон равны 14 и 8. Найдите основания трапеции.

Answer 1

Трапецию можно достроить до треугольника)))
если сумма острых углов трапеции = 90°, то 
трапеция достраивается до прямоугольного треугольника.
для прямоугольного треугольника известно:
медиана к гипотенузе равна половине гипотенузы)))
это радиус описанной около прямоугольного треугольника окружности, 
центр которой - середина гипотенузы
(т.к. прямой угол опирается на диаметр описанной окружности)))
это основа для вычислений...
и еще: средняя линия трапеции = полусумме длин ее оснований.

Answer 2

Finding the Bases of the Trapezoid

To find the bases of the trapezoid, we can use the given information about the angles and the lengths of the segments connecting the midpoints of the opposite sides.

Let's denote the bases of the trapezoid as AB and CD. We are given that the angles at one of the bases are 42° and 48°. Let's assume that these angles are at base AB.

Since the angles at the base AB are 42° and 48°, the angles at the other base CD will be supplementary to these angles. Therefore, the angles at base CD will be 138° (180° - 42°) and 132° (180° - 48°).

Now, let's consider the segments connecting the midpoints of the opposite sides. We are given that these segments are 14 and 8 units long. Let's denote the midpoint of side AD as point M and the midpoint of side BC as point N.

Using the properties of trapezoids, we know that the segments connecting the midpoints of the opposite sides are parallel to the bases and their lengths are half the lengths of the bases. Therefore, we can set up the following equations:

MN = 14 (length of the segment connecting the midpoints of the opposite sides)

AB = 2 * MN (length of base AB)

CD = 2 * MN (length of base CD)

Substituting the value of MN, we can find the lengths of the bases:

AB = 2 * 14 = 28

CD = 2 * 14 = 28

Therefore, the lengths of the bases of the trapezoid are both 28 units.

Please note that the search results provided did not contain the specific information required to solve this problem. The solution was derived using the properties of trapezoids and basic geometry principles.