Question

В равнобедренной трапеции основания равны 7см и 23 см.Высота трапеции 15 см .Найти периметр и среднюю линию

Answer 1

Средняя линия трапеции равна (7+23)=30 см.
ΔАВК=ΔDСМ, АК=МD=(23-7)/2=8 см.
 ΔАВК. АВ²=АК²+ВК², АВ²=225+64=289, АВ=√289=17 см. Р(АВСD)=7+23+17+17=64 см.
 Ответ: 64 см, 30 см.

Answer 2

Given Information:

We are given that the trapezoid is isosceles, with base lengths of 7 cm and 23 cm, and a height of 15 cm.

Perimeter of the Trapezoid:

To find the perimeter of the trapezoid, we need to add up the lengths of all its sides.

The formula for the perimeter of a trapezoid is: Perimeter = sum of all sides

In this case, the trapezoid has two parallel sides (bases) and two non-parallel sides (legs).

The length of the parallel sides (bases) can be found by adding the lengths of the two bases: Length of the bases = 7 cm + 23 cm

The length of the non-parallel sides (legs) can be found using the Pythagorean theorem. Since the trapezoid is isosceles, the legs are congruent.

Let's denote the length of each leg as x.

Using the Pythagorean theorem, we have: x^2 + (7 cm - 23 cm/2)^2 = 15 cm^2

Simplifying the equation, we get: x^2 + 49 cm^2 - 2 * 7 cm * 23 cm/2 + (23 cm/2)^2 = 15 cm^2

Simplifying further, we have: x^2 + 49 cm^2 - 7 cm * 23 cm + (23 cm/2)^2 = 15 cm^2

Simplifying again, we get: x^2 + 49 cm^2 - 161 cm^2 + 529 cm^2/4 = 15 cm^2

Combining like terms, we have: x^2 - 112 cm^2 + 529 cm^2/4 = 15 cm^2

Simplifying further, we get: x^2 + 529 cm^2/4 - 112 cm^2 = 15 cm^2

Combining like terms, we have: x^2 + 529 cm^2/4 - 112 cm^2 = 15 cm^2

Simplifying again, we get: x^2 + 529 cm^2/4 - 448 cm^2/4 = 15 cm^2

Combining like terms, we have: x^2 + 81 cm^2/4 = 15 cm^2

Simplifying further, we get: x^2 + 81 cm^2/4 = 15 cm^2

Subtracting 81 cm^2/4 from both sides, we have: x^2 = 15 cm^2 - 81 cm^2/4

Simplifying, we get: x^2 = 60 cm^2 - 81 cm^2/4

Multiplying both sides by 4, we have: 4x^2 = 240 cm^2 - 81 cm^2

Simplifying further, we get: 4x^2 = 159 cm^2

Taking the square root of both sides, we have: 2x = √(159 cm^2)

Simplifying, we get: 2x = 12.61 cm

Dividing both sides by 2, we have: x = 6.305 cm

Now that we have the length of each leg, we can find the perimeter of the trapezoid by adding up the lengths of all its sides: Perimeter = 7 cm + 23 cm + 6.305 cm + 6.305 cm

Calculating the perimeter, we get: Perimeter = 42.61 cm

Therefore, the perimeter of the trapezoid is approximately 42.61 cm.

Midline of the Trapezoid:

The midline of a trapezoid is the line segment that connects the midpoints of the two non-parallel sides (legs).

To find the length of the midline, we need to find the average of the lengths of the two non-parallel sides.

Let's denote the length of each leg as x.

Using the Pythagorean theorem, we have: x^2 + (7 cm - 23 cm/2)^2 = 15 cm^2

We have already solved this equation earlier and found that x = 6.305 cm.

The length of the midline can be found by taking the average of the lengths of the two legs: Midline = (7 cm + 6.305 cm)/2

Calculating the midline, we get: Midline = 6.6525 cm

Therefore, the length of the midline of the trapezoid is approximately 6.6525 cm.

Summary:

- The perimeter of the trapezoid is approximately 42.61 cm. - The length of the midline of the trapezoid is approximately 6.6525 cm.

Please note that the calculations provided are approximate due to rounding.