Question

диагональ прямоугольника равна 20 см. угол между диагоналями равен 60 градусов. вычислить длины дуг, на которые делят вершины прямоуг-ка описанную около него окружность. помогите!!!!!!!!!!!

Answer 1

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Answer 2

Given Information:

We are given the following information about a rectangle: - The diagonal of the rectangle is 20 cm. - The angle between the diagonals is 60 degrees.

Solution:

To find the lengths of the arcs that the vertices of the rectangle divide on the circle circumscribed around it, we need to calculate the lengths of the sides of the rectangle.

Let's denote the sides of the rectangle as follows: - Length of the longer side: a - Length of the shorter side: b

Using the given information, we can find the lengths of the sides of the rectangle.

Finding the Lengths of the Sides:

We can use the properties of a rectangle to find the lengths of its sides.

1. The diagonal of a rectangle divides it into two congruent right triangles. 2. The angle between the diagonals is 60 degrees.

Using these properties, we can apply trigonometry to find the lengths of the sides.

Let's consider one of the right triangles formed by the diagonal and one side of the rectangle.

In the right triangle: - The hypotenuse is the diagonal of the rectangle, which is 20 cm. - One of the acute angles is 60 degrees.

Using the trigonometric ratios, we can find the lengths of the sides of the right triangle.

Let's denote the length of the longer side of the rectangle as a and the length of the shorter side as b.

Using the sine function, we have: sin(60 degrees) = a / 20

Simplifying the equation, we get: a = 20 * sin(60 degrees)

Similarly, using the cosine function, we have: cos(60 degrees) = b / 20

Simplifying the equation, we get: b = 20 * cos(60 degrees)

Now, we can calculate the values of a and b.

Calculating the Lengths of the Sides:

Using the trigonometric functions, we can calculate the lengths of the sides of the rectangle.

Using a calculator, we find: - a ≈ 17.32 cm - b ≈ 10 cm

Now that we have the lengths of the sides of the rectangle, we can proceed to calculate the lengths of the arcs.

Calculating the Lengths of the Arcs:

To calculate the lengths of the arcs that the vertices of the rectangle divide on the circle circumscribed around it, we need to find the radius of the circle.

The radius of the circle can be calculated using the formula: radius = diagonal / 2

Substituting the given value, we have: radius = 20 cm / 2 = 10 cm

Now, we can calculate the lengths of the arcs using the formula: arc length = radius * angle in radians

Since the angle between the diagonals is 60 degrees, we need to convert it to radians.

Using the formula: angle in radians = angle in degrees * (π / 180)

Substituting the given value, we have: angle in radians = 60 degrees * (π / 180) ≈ 1.047 radians

Now, we can calculate the lengths of the arcs.

Using the formula: arc length = radius * angle in radians

Substituting the values, we have: - Length of the arc at the longer side of the rectangle: arc length = 10 cm * 1.047 radians ≈ 10.47 cm - Length of the arc at the shorter side of the rectangle: arc length = 10 cm * 1.047 radians ≈ 10.47 cm

Therefore, the lengths of the arcs that the vertices of the rectangle divide on the circle circumscribed around it are approximately 10.47 cm each.

Summary:

- The lengths of the sides of the rectangle are approximately 17.32 cm and 10 cm. - The lengths of the arcs that the vertices of the rectangle divide on the circle circumscribed around it are approximately 10.47 cm each.

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