Question

Прямая параллельная основанию треугольника делит его на треугольник и четырехугольник, площади которых относятся как 1:8. Найдите периметр большего треугольника, если периметр меньшего равен 7 см. Срочно надо! И подробно!

Answer 1

Площадь АВС=1+8=9
Площади подобных треугольников относятся как квадрат коэффициента пропорциональности:
Площадь АВС к площади DBE = 9 к 1
Т.е $( \frac{3}{1} )^2$
Коэффициент равен 3
Периметр умножаем на коэф.  подобия
7*3=21

Ответ:21

Answer 2

Problem Analysis

We are given that a line parallel to the base of a triangle divides it into a smaller triangle and a quadrilateral, and the areas of these two figures are in the ratio of 1:8. We are also given that the perimeter of the smaller triangle is 7 cm. We need to find the perimeter of the larger triangle.

Solution

Let's denote the sides of the smaller triangle as a, b, and c, and the sides of the larger triangle as A, B, and C. We know that the perimeter of the smaller triangle is 7 cm, so a + b + c = 7.

Since the areas of the smaller triangle and the quadrilateral are in the ratio of 1:8, we can write the following equation:

Area of smaller triangle / Area of quadrilateral = 1 / 8

The area of a triangle can be calculated using the formula Area = (1/2) * base * height. Since the line is parallel to the base, the height of both the smaller and larger triangles will be the same. Therefore, we can write:

(1/2) * base of smaller triangle * height = (1/2) * base of larger triangle * height / 8

Simplifying this equation, we get:

base of smaller triangle / base of larger triangle = 1 / 8

Since the line is parallel to the base, the corresponding sides of the smaller and larger triangles are proportional. Therefore, we can write:

a / A = b / B = c / C = 1 / 8

Now, we have three equations:

1. a + b + c = 7 2. a / A = b / B = c / C = 1 / 8

We can solve these equations to find the values of A, B, and C.

Solving the Equations

From equation 2, we can write:

a = (1/8) * A b = (1/8) * B c = (1/8) * C

Substituting these values into equation 1, we get:

(1/8) * A + (1/8) * B + (1/8) * C = 7

Multiplying both sides of the equation by 8, we get:

A + B + C = 56

Therefore, the perimeter of the larger triangle is 56 cm.

Answer

The perimeter of the larger triangle is 56 cm.

Verification

Let's verify our answer using the given information. We know that the perimeter of the smaller triangle is 7 cm. Let's calculate the perimeter of the larger triangle using the values we found.

Using the values we found earlier:

a = (1/8) * A b = (1/8) * B c = (1/8) * C

Substituting these values into the equation a + b + c = 7, we get:

(1/8) * A + (1/8) * B + (1/8) * C = 7

Multiplying both sides of the equation by 8, we get:

A + B + C = 56

Therefore, our answer is verified.

Conclusion

The perimeter of the larger triangle is 56 cm.