Question

ПОМОГИТЕ ПОЖАЛУЙСТА(ЖЕЛАТЕЛЬНО ФОТО) 4. Периметр трапеции равен 50 см, а сумма непараллельных сторон равна 20 см. Найдите среднюю линию трапеции.​

Answer 1

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Answer 2

Perimeter of a Trapezoid

To find the length of the median (also known as the average line or midline) of a trapezoid, we need to know the perimeter of the trapezoid and the sum of the lengths of the non-parallel sides.

Given: - Perimeter of the trapezoid = 50 cm - Sum of the lengths of the non-parallel sides = 20 cm

Let's denote the lengths of the parallel sides of the trapezoid as a and b, and the lengths of the non-parallel sides as c and d.

The perimeter of a trapezoid is given by the formula: Perimeter = a + b + c + d

We are given that the perimeter is 50 cm, so we can write the equation: 50 = a + b + c + d We are also given that the sum of the lengths of the non-parallel sides is 20 cm, so we can write the equation: c + d = 20 To find the length of the median (average line or midline), we need to find the difference between the lengths of the parallel sides and divide it by 2.

Let's denote the length of the median as m.

The length of the median can be calculated using the formula: m = (|a - b|) / 2

Now, let's solve the equations to find the lengths of the sides and the median of the trapezoid.

Solution:

From equation we can rewrite it as: a + b = 50 - (c + d)

Substituting the value of c + d from the second equation, we get: a + b = 50 - 20 a + b = 30

To find the length of the median, we need to find the difference between the lengths of the parallel sides, which can be written as: |a - b| = 2m

Substituting the value of a + b from the previous equation, we get: |a - b| = 2m |30 - b| = 2m

Since the median divides the trapezoid into two equal areas, the lengths of the parallel sides are equal. Therefore, we can simplify the equation as: |30 - b| = 2m |30 - b| = 2m

To find the value of m, we need to consider two cases:

Case 1: When 30 - b ≥ 0 In this case, we have: 30 - b = 2m b = 30 - 2m

Case 2: When 30 - b < 0 In this case, we have: b - 30 = 2m b = 30 + 2m

Now, we have two equations for the length of the parallel side b. We can substitute these values into the equation a + b = 30 to find the values of a.

Substituting b = 30 - 2m into a + b = 30, we get: a + (30 - 2m) = 30 a = 2m

Substituting b = 30 + 2m into a + b = 30, we get: (30 + 2m) + b = 30 a = -2m

Since the length of a side cannot be negative, we can disregard the solution a = -2m.

Therefore, the length of the median is given by: m = (|a - b|) / 2 m = (|2m - (30 - 2m)|) / 2 m = (|4m - 30|) / 2

To find the value of m, we need to consider two cases:

Case 1: When 4m - 30 ≥ 0 In this case, we have: 4m - 30 = 2m 2m = 30 m = 15

Case 2: When 4m - 30 < 0 In this case, we have: -(4m - 30) = 2m -4m + 30 = 2m 6m = 30 m = 5

Therefore, the length of the median of the trapezoid can be either 15 cm or 5 cm.

Please note that without additional information about the trapezoid, such as the lengths of the parallel sides or the angles, we cannot determine the exact length of the median.