Question

Средняя линия трапеции равна 16 см,а периметр 64 см.Докажите что в данную трапецию можно вписать окружность

Answer 1

Вот т. к. окружность вписана в трапецию, то её 2 радиуса являются средней линией.
средняя линия = 5+5 = 10
(а+в) /2 = 10
а+в = 20
периметр трапеции = 20+16+16 = 52 см
а и в это основания трапеции

Answer 2

Given Information

We are given that the average length of the bases of a trapezoid is 16 cm and the perimeter is 64 cm.

Proof that a Circle can be Inscribed in the Trapezoid

To prove that a circle can be inscribed in the given trapezoid, we need to show that the perpendicular bisectors of the trapezoid's sides intersect at a single point, which is the center of the inscribed circle.

Let's denote the lengths of the trapezoid's bases as a and b. According to the given information, the average length of the bases is 16 cm. Therefore, we can write the equation:

(a + b) / 2 = 16 Simplifying the equation, we have:

a + b = 32

The perimeter of the trapezoid is given as 64 cm. The perimeter of a trapezoid can be calculated by adding the lengths of all its sides. In this case, we have:

a + b + 2x + 2y = 64

where x and y represent the lengths of the trapezoid's non-parallel sides.

Since the trapezoid has two pairs of congruent sides, we can write:

x = y

Substituting this into the perimeter equation, we get:

a + b + 4x = 64

Rearranging the equation, we have:

a + b = 64 - 4x

Since we already know that a + b = 32, we can equate the two expressions:

32 = 64 - 4x

Simplifying the equation, we find:

4x = 32

Dividing both sides by 4, we get:

x = 8

Since x represents the length of one of the non-parallel sides, we can conclude that both non-parallel sides have a length of 8 cm.

Now, let's consider the perpendicular bisectors of the trapezoid's sides. The perpendicular bisector of a line segment is a line that intersects the segment at its midpoint and is perpendicular to it.

Since the trapezoid has two pairs of congruent sides, the perpendicular bisectors of these sides will intersect at the midpoint of the bases. This midpoint is also the center of the inscribed circle.

Therefore, we have shown that the perpendicular bisectors of the trapezoid's sides intersect at a single point, which is the center of the inscribed circle. Hence, a circle can be inscribed in the given trapezoid.

Note: The lengths of the non-parallel sides of the trapezoid are not explicitly given in the problem statement. However, we have deduced that both non-parallel sides have a length of 8 cm based on the given information and the properties of a trapezoid.