Question

В прямоугольной трапеции 1 из углов равен 60 градусам,а большая боковая сторона равна 8 см, найти ВС и АД и радиус вписаной окружности

Answer 1

В прямоугольной трапеции один из углов равен 60 ,а большая боковая сторона равна 8 см. Найти ВС и АД и радиус вписаной окружности
Решение.
См. рисунок 1.
Проведем высоту СК.
В прямоугольном треугольнике  CKD  катет КD  равен половине гипотенузы, так как лежит против угла в 30°
KD = 4 см.
 Тогда по теореме Пифагора СК²=СD² - KD²= 8²-4²=64-16=48
CK=4√3 см.
По свойству четырехугольника, описанного около окужности, суммы противоположных сторон равны
АВ + CD = ВC + AD
Значит  ВС + AD = 4√3 + 8
Но  так как  BC = AK    и     AD = АК + KD = ВС + KD,
то    ВС + ВС + 4 = 4 √3 + 8    ⇒  2 ВС = 4√3 + 4  ⇒  ВС = 2√3 + 2
AD = BC + KD = 2√3 + 2 + 4 = 2 √3 + 6

r = CK/2 = 4√3/2 = 2√3
Ответ. верхнее основание 2√3 + 2, нижнее основание 2 √3 + 6, радиус вписанной окружности
2√3

Answer 2

Problem Analysis

We are given a trapezoid with one angle measuring 60 degrees and the length of the longer base equal to 8 cm. We need to find the lengths of sides BC and AD, as well as the radius of the inscribed circle.

Solution

To solve this problem, we can use the properties of a trapezoid and trigonometry.

Let's label the trapezoid as ABCD, where AB is the longer base, BC is one of the legs, and AD is the other leg. Let's also label the center of the inscribed circle as O, and the points where the circle touches the trapezoid as A1, B1, C1, and D1.

Finding the Length of Side BC

Since one of the angles of the trapezoid is 60 degrees, we can use trigonometry to find the length of side BC.

Let's consider triangle ABC. We know that angle BAC is 60 degrees and side AB is 8 cm. Using the trigonometric function cosine, we can find the length of side BC.

cos(60) = BC / AB

Simplifying the equation, we get:

BC = AB * cos(60)

Substituting the values, we have:

BC = 8 cm * cos(60)

Using a calculator, we find that cos(60) is 0.5. Therefore:

BC = 8 cm * 0.5 = 4 cm.

So, the length of side BC is 4 cm.

Finding the Length of Side AD

To find the length of side AD, we can use the fact that the opposite sides of a trapezoid are parallel and equal in length.

Since side BC is parallel to side AD, we can conclude that the length of side AD is also 4 cm.

Therefore, the length of side AD is 4 cm.

Finding the Radius of the Inscribed Circle

To find the radius of the inscribed circle, we can use the formula:

Radius = Area / Semiperimeter.

The area of a trapezoid can be calculated using the formula:

Area = (sum of the lengths of the bases) * height / 2.

In this case, the height of the trapezoid is the perpendicular distance between the bases. Let's label this height as h.

The sum of the lengths of the bases is AB + CD = 8 cm + 4 cm = 12 cm.

Therefore, the area of the trapezoid is:

Area = (12 cm) * h / 2.

The semiperimeter of the trapezoid can be calculated using the formula:

Semiperimeter = (sum of the lengths of the bases) / 2.

In this case, the semiperimeter is:

Semiperimeter = (8 cm + 4 cm) / 2 = 6 cm.

Now, let's substitute the values into the formula for the radius:

Radius = (12 cm * h / 2) / 6 cm.

Simplifying the equation, we get:

Radius = h / 2.

To find the value of h, we can use the Pythagorean theorem.

Let's consider triangle ABC. We know that angle BAC is 60 degrees, side AB is 8 cm, and side BC is 4 cm. Using the trigonometric function sine, we can find the value of h.

sin(60) = h / BC.

Simplifying the equation, we get:

h = BC * sin(60).

Substituting the values, we have:

h = 4 cm * sin(60).

Using a calculator, we find that sin(60) is approximately 0.866. Therefore:

h = 4 cm * 0.866 = 3.464 cm.

Now, let's substitute the value of h into the formula for the radius:

Radius = 3.464 cm / 2 = 1.732 cm.

Therefore, the radius of the inscribed circle is approximately 1.732 cm.

Summary

In summary, for the given trapezoid with one angle measuring 60 degrees and the length of the longer base equal to 8 cm, we have found the following measurements: - Length of side BC: 4 cm. - Length of side AD: 4 cm. - Radius of the inscribed circle: approximately 1.732 cm.

Please let me know if there is anything else I can help you with!