Question

Прямая,проведенная параллельно боковой стороне трапеции через конец меньшего основания, равного 7 см, отсекает треугольник, периметр которого равен 18 см. Найдите периметр трапеции.

Answer 1

Трапеция АВСД, ВК параллельноСД, КВСД - параллелограмм, ВК=СД=7, ВС=КД, периметр треугольника АВК =АВ+ВК(СД)+АК=18

периметр трапеции = АВ+ВС+СД(ВК)+КД(ВС)+АК = 18 +2ВС=18+14=32

Answer 2

Problem Analysis

We are given a trapezoid and a line parallel to one of its lateral sides. This line intersects the trapezoid at the end of the smaller base, which is 7 cm long. The triangle formed by this line and the trapezoid has a perimeter of 18 cm. We need to find the perimeter of the trapezoid.

Solution

Let's denote the trapezoid as ABCD, with AB as the longer base, CD as the shorter base, and AD and BC as the lateral sides. Let M be the point where the line intersects the trapezoid, with M on AD and N on BC.

To find the perimeter of the trapezoid, we need to find the lengths of the sides AD, AB, BC, and CD.

Since the line is parallel to the lateral side BC, we can conclude that triangles AMN and ABC are similar. This means that the ratio of the corresponding sides of these triangles is equal. Therefore, we have:

AM/AB = AN/AC = MN/BC

Let's denote the length of MN as x. Since the perimeter of triangle AMN is given as 18 cm, we can write the following equation:

AM + AN + MN = 18

Substituting the ratios from above, we can express AM and AN in terms of x:

AM = (AB/AC) * x AN = (BC/AC) * x

Substituting these expressions into the equation, we get:

(AB/AC) * x + (BC/AC) * x + x = 18

Simplifying the equation, we have:

x * (AB/AC + BC/AC + 1) = 18

Now, let's find the lengths of AB, AC, and BC. We are given that CD is 7 cm, but we need to find the lengths of the other sides.

To find AB, we can use the fact that triangles ABC and ACD are similar. This gives us the following ratio:

AB/AC = BC/CD

Substituting the values, we have:

AB/AC = BC/7

To find BC, we can use the fact that triangles ABC and BCD are similar. This gives us the following ratio:

BC/AB = CD/BD

Substituting the values, we have:

BC/AB = 7/BD

Since AB + BD = AD, we can express BD in terms of AB and AD:

BD = AD - AB

Substituting this into the equation, we have:

BC/AB = 7/(AD - AB)

Now, let's substitute these expressions into the equation we derived earlier:

x * (AB/AC + BC/AC + 1) = 18

Substituting AB/AC and BC/AB, we have:

x * (BC/7 + 7/(AD - AB) + 1) = 18

Simplifying the equation, we have:

x * (BC(AD - AB) + 7(AD - AB) + 7BC) = 18 * 7 * (AD - AB)

Expanding and rearranging the equation, we get:

x * (BC * AD - BC * AB + 7 * AD - 7 * AB + 7 * BC) = 126 * (AD - AB)

Now, let's simplify the equation further:

x * (BC * AD + 7 * AD + 7 * BC) = 126 * (AD - AB) + x * (BC * AB + 7 * AB)

Dividing both sides of the equation by (AD - AB + x), we get:

x * (BC * AD + 7 * AD + 7 * BC) / (AD - AB + x) = 126 + x * (BC * AB + 7 * AB) / (AD - AB + x)

Now, let's substitute the given values and solve for x:

x * (BC * AD + 7 * AD + 7 * BC) / (AD - AB + x) = 126 + x * (BC * AB + 7 * AB) / (AD - AB + x)

Substituting BC = 7, CD = 7, and the given perimeter of the triangle as 18, we have:

x * (7 * AD + 7 * AD + 7 * 7) / (AD - AB + x) = 126 + x * (7 * AB + 7 * AB) / (AD - AB + x)

Simplifying the equation, we have:

x * (14 * AD + 49) / (AD - AB + x) = 126 + x * (14 * AB) / (AD - AB + x)

Now, let's solve this equation to find the value of x.