Question

Боковые стороны равнобедренной трапеции продолжены до пересечения в точке M . Основания трапеции равны 3 см и 9 см, боковая сторона равна 8 см. Найти расстояние от точки M до конца меньшего основания

Answer 1

Ответ:

4 см

Пошаговое объяснение:

Рассмотрим приложенные рисунки.

Опустим высоты из точки М на меньшее основание и из точки В на большее основание. Получаем два прямоугольных треугольника. У этих треугольников равны все углы, значит треугольники подобны.

Очевидно, что ВО=ВС÷2=1,5 и АК=АD÷2-ВО=3

Тогда из свойств подобия треугольников получаем коэффициент подобия: к=$\frac{AK}{BO}$=3÷1,5=2.

Значит искомая длина ВМ=АВ÷2=8÷2=4 см

Answer 2

Ответ:

4 см.

Пошаговое объяснение:

Answer 3

Given Information:

We are given a trapezoid with equal legs. The lengths of the bases are 3 cm and 9 cm, and one of the legs is 8 cm. We need to find the distance from point M to the end of the smaller base.

Solution:

To find the distance from point M to the end of the smaller base, we can use the properties of similar triangles.

Let's label the trapezoid as ABCD, with AB as the larger base, CD as the smaller base, and AD and BC as the legs. Let E be the point where the extension of BC intersects the extension of AD, and let F be the point where the extension of AB intersects the extension of CD.

Since the trapezoid is isosceles, we know that AD = BC. Let's denote this length as x.

Using the properties of similar triangles, we can set up the following proportion:

BC / AD = CF / AF

Substituting the given values, we have:

8 / x = (x + 9) / 3

Cross-multiplying, we get:

8 * 3 = (x + 9) * x

Simplifying the equation, we have:

24 = x^2 + 9x

Rearranging the equation, we get:

x^2 + 9x - 24 = 0

Now, we can solve this quadratic equation to find the value of x.

Using the quadratic formula, we have:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = 9, and c = -24.

Substituting the values, we get:

x = (-9 ± √(9^2 - 4 * 1 * -24)) / (2 * 1)

Simplifying further, we have:

x = (-9 ± √(81 + 96)) / 2

x = (-9 ± √177) / 2

Since the length of a side cannot be negative, we can ignore the negative solution.

Therefore, the length of the legs of the trapezoid is:

x = (-9 + √177) / 2

Now, we can find the distance from point M to the end of the smaller base, which is CD.

Since the trapezoid is isosceles, we know that the distance from M to the end of the smaller base is equal to the distance from M to the end of the larger base.

Therefore, the distance from point M to the end of the smaller base is:

(-9 + √177) / 2 cm.

Please note that the above solution assumes that point M lies on the extension of the legs of the trapezoid. If point M is located inside the trapezoid, the solution may be different.