Question

Диагональ равнобедренной трапеции перпендикулярна боковой стороне. найдите радиус окружности, описанной около трапеции, если диагональ равна 12 см, а боковая сторона - 9 см. Заранее огромное спасибо

Answer 1

АВСД трапеция

АВ=СД=9

ВД=АС=12

АД=√144+81=15

СА*ВД=АВ*СД+АД*ВС

ВС=(144-81)/15=4,2

т.О пересечение оси симметрии трапеции и диагонали

ВО/ОД=ВС/АД (по теорем Фалеса)

ВО+ОД=15  ⇒ОД=15-ОВ

 

ВО/15-ВО=4,2/15

ВО=63/19,2=3,28

т.Р пересечение оси симметрии трапеции и серединного перпендикуляра АВ, Е середина АВ

ЕР=ВО=3,28

R²=АЕ²+ЕР²=4,5²+3,28²=31,02

R=5.57 см

Answer 2

Finding the Radius of the Circumscribed Circle

To find the radius of the circle circumscribed around a trapezoid, we can use the properties of a trapezoid and the Pythagorean theorem.

Given: - Diagonal of the trapezoid = 12 cm - Length of one of the lateral sides (a leg) = 9 cm

To find the radius of the circumscribed circle, we can follow these steps:

1. Draw the trapezoid and label the given measurements. 2. Draw the diagonal perpendicular to one of the lateral sides. 3. Divide the trapezoid into two right triangles by drawing a perpendicular from one of the vertices to the base. 4. Use the Pythagorean theorem to find the length of the other leg of the right triangle. 5. Since the diagonal is perpendicular to the base, the length of the other leg is equal to the height of the trapezoid. 6. Use the height to find the radius of the circumscribed circle.

Let's calculate the radius using the given measurements.

Calculation:

In the right triangle formed by the diagonal and the height of the trapezoid, we can use the Pythagorean theorem:

a^2 + b^2 = c^2

Where: - a = length of one leg (9 cm) - b = length of the other leg (height of the trapezoid) - c = length of the hypotenuse (diagonal of the trapezoid) = 12 cm

Substituting the given values into the equation:

9^2 + b^2 = 12^2

Simplifying the equation:

81 + b^2 = 144

b^2 = 144 - 81

b^2 = 63

Taking the square root of both sides:

b = √63

Calculating the value of √63:

b ≈ 7.937 cm

Now that we have the height of the trapezoid, we can find the radius of the circumscribed circle.

In a trapezoid, the height is perpendicular to the base and passes through the midpoint of the diagonal. Therefore, the height is also the radius of the circumscribed circle.

So, the radius of the circumscribed circle is approximately 7.937 cm.

Please note that the calculations are approximate due to rounding.

I hope this helps! Let me know if you have any further questions.