Question

Разность оснований трапеции равна 4 см, а средняя линия равна 10 см. Найдите основания трапеции.

Answer 1

Обозначим большее основание как а, а меньшее как b. Учтем при этом следующее:
Средняя линия трапеции параллельна основаниям трапеции и равна их полу-сумме.
Значит:
a - b = 4
(a + b) / 2 = 10
Откуда a = b + 4, тогда:
(b + 4 + b):2 = 10
2b + 4 = 20
b = 8
Следовательно: a = b + 4 = 12.

Ответ: основания трапеции равны 8 и 12 сантиметров.

Answer 2

Finding the Bases of a Trapezoid

To find the bases of a trapezoid, we are given that the difference between the bases is 4 cm and the length of the median (or average) line is 10 cm.

Let's denote the lengths of the bases as b1 and b2.

We can set up the following equation based on the given information:

b2 - b1 = 4 (Equation 1)

We also know that the median line is the average of the bases. The formula for the median line of a trapezoid is:

Median = (b1 + b2) / 2 (Equation 2)

Substituting the value of the median line (10 cm) into Equation 2, we get:

(b1 + b2) / 2 = 10

Simplifying Equation 2, we have:

b1 + b2 = 20 (Equation 3)

Now we have a system of two equations (Equations 1 and 3) with two unknowns (b1 and b2). We can solve this system to find the values of the bases.

To solve the system, we can use the method of substitution. Rearranging Equation 3, we get:

b1 = 20 - b2

Substituting this value of b1 into Equation 1, we have:

(20 - b2) - b2 = 4

Simplifying the equation:

20 - 2b2 = 4

-2b2 = 4 - 20

-2b2 = -16

Dividing both sides by -2:

b2 = 8

Substituting the value of b2 back into Equation 3, we can find b1:

b1 + 8 = 20

b1 = 20 - 8

b1 = 12

Therefore, the bases of the trapezoid are 12 cm and 8 cm.