Question

Диагональ равнобедренной трапеции с основаниями 8 см, и 5 см, является биссектрисой острого угла трапеции.Найдите периметр трапеции.

Answer 1

Рисуем трапецию АВСД. Проводим диагональ АС. 
АД = 8 см 
ВС = 5 см 
Угол ВАС = угол САД (по условию) 
Угол ВСА = угол САД (по свойствам параллельных прямых) 
Следвательно, 
угол ВАС = угол ВСА и треугольник АВС - равнобедренный 
СД = АВ = ВС = 5 см 
Периметр 
Р = АВ + ВС + СД + АД = 5 + 5 + 5 + 8 = 23 см

Answer 2

Diagonal of an Isosceles Trapezoid

To find the perimeter of a trapezoid, we need to know the lengths of its bases and its two non-parallel sides. However, in this case, we are given the lengths of the bases and the information that the diagonal is the bisector of one of the acute angles of the trapezoid.

Let's denote the lengths of the bases as a (8 cm) and b (5 cm). We are also given that the diagonal is the bisector of one of the acute angles.

To find the perimeter, we need to determine the lengths of the non-parallel sides. Since the diagonal is the bisector of one of the acute angles, it divides the trapezoid into two congruent right triangles. Let's call the length of the diagonal d.

Using the properties of right triangles, we can apply the Pythagorean theorem to find the length of the diagonal d. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this case, the diagonal d is the hypotenuse, and the two sides of the right triangle are the lengths of the bases a and b. Therefore, we have:

d^2 = a^2 + b^2.

Let's substitute the given values:

d^2 = 8^2 + 5^2.

Simplifying the equation:

d^2 = 64 + 25.

d^2 = 89.

Taking the square root of both sides:

d = √89.

Now that we have the length of the diagonal d, we can calculate the perimeter of the trapezoid. The perimeter is the sum of all the sides of the trapezoid.

The perimeter of a trapezoid is given by the formula:

Perimeter = a + b + c + d,

where c and d are the non-parallel sides.

In this case, since the diagonal is the bisector of one of the acute angles, the non-parallel sides are congruent. Therefore, we have:

Perimeter = a + b + 2c.

Substituting the given values:

Perimeter = 8 + 5 + 2√89.

Simplifying the equation:

Perimeter = 13 + 2√89.

Therefore, the perimeter of the trapezoid is 13 + 2√89 cm.

Please note that the above calculations are based on the given information and assumptions about the properties of the trapezoid. If there are any additional constraints or information, please let me know, and I will be happy to assist you further.

Answer:

The perimeter of the trapezoid is 13 + 2√89 cm.