Question

Сторона прямокутника вдвічі менша від діагоналей. Знайдіть кут між діагоналями прямокутника

Answer 1

Сторона що лежить проти кута 30 градусів дорівнює половині гіпотенузи . Отже сторона це катет ,діагональ це гіпотенуза отже кут між ними дорівнює 30 градусів

Answer 2

Finding the Angle Between the Diagonals of a Rectangle

To find the angle between the diagonals of a rectangle, we need to use the given information that one side of the rectangle is half the length of the diagonals.

Let's assume that the length of one side of the rectangle is x. Since the other side is twice as long as this, the length of the other side is 2x.

Now, let's consider the diagonals of the rectangle. The diagonals of a rectangle are congruent, meaning they have the same length. Let's denote the length of the diagonals as d.

Using the Pythagorean theorem, we can relate the sides and the diagonals of the rectangle:

d^2 = x^2 + (2x)^2

Simplifying this equation, we get:

d^2 = x^2 + 4x^2

d^2 = 5x^2

Taking the square root of both sides, we find:

d = sqrt(5x^2)

Now, let's find the angle between the diagonals. We can use the cosine formula to calculate the angle:

cos(angle) = (x^2 + x^2 - d^2) / (2x * x)

Simplifying this equation, we get:

cos(angle) = (2x^2 - d^2) / (2x^2)

cos(angle) = (2x^2 - 5x^2) / (2x^2)

cos(angle) = -3x^2 / (2x^2)

cos(angle) = -3/2

To find the angle, we can take the inverse cosine (arccos) of -3/2:

angle = arccos(-3/2)

Using a calculator, we find that the angle is approximately 131.81 degrees.

Therefore, the angle between the diagonals of the rectangle is approximately 131.81 degrees.

Please note that the above calculations assume that the rectangle is a right-angled rectangle (a rectangle with 90-degree angles). If the rectangle is not right-angled, the angle between the diagonals will be different.