Question

У прямокутній трапеції гострий кут дорівнює 30°, а менша діагональ ділить прямий кут навпіл. Знайдіть бічні сторони трапеції, якщо її менша основа дорівнює 4 см. У відповідь запишіть значення меншої бічної сторони. Срочно пожалуйста даю 60 баллов

Answer 1

Ответ:

4 і 8 см

Пошаговое объяснение:

Answer 2

Problem Analysis

We are given a trapezoid with an acute angle of 30° and the smaller diagonal dividing the right angle in half. We need to find the lengths of the lateral sides of the trapezoid, given that the length of the smaller base is 4 cm.

Solution

Let's denote the lengths of the lateral sides of the trapezoid as a and b. We can use the given information to set up equations and solve for the unknowns.

From the given information, we know that the acute angle of the trapezoid is 30° and the smaller diagonal divides the right angle in half. Let's consider the right triangle formed by the smaller diagonal, the smaller base, and one of the lateral sides. In this triangle, the acute angle is 30° and the right angle is divided in half, so the other acute angle is also 30°.

Using trigonometric ratios, we can relate the lengths of the sides of this right triangle. Let's consider the side adjacent to the acute angle (the smaller base) as the base of the triangle, the side opposite the acute angle (the lateral side) as the height of the triangle, and the hypotenuse (the smaller diagonal) as the hypotenuse of the triangle.

We can use the tangent function to relate the lengths of the sides:

tan(30°) = height / base

Since the base is given as 4 cm, we can substitute the values and solve for the height:

tan(30°) = height / 4

Using the value of the tangent of 30° (which is approximately 0.577), we can solve for the height:

0.577 = height / 4

height = 0.577 * 4

height = 2.308 cm

Therefore, the length of the smaller lateral side of the trapezoid is 2.308 cm.

To find the length of the larger lateral side, we can use the fact that the smaller diagonal divides the right angle in half. This means that the right triangle formed by the larger diagonal, the smaller diagonal, and the larger lateral side is also a right triangle with a right angle and an acute angle of 30°.

Using the same trigonometric reasoning as before, we can set up the following equation:

tan(30°) = height / larger_base

Since the height is the same as before (2.308 cm) and the larger base is unknown, we can solve for the larger base:

0.577 = 2.308 / larger_base

Simplifying the equation, we find:

larger_base = 2.308 / 0.577

larger_base ≈ 3.999 cm

Therefore, the length of the larger lateral side of the trapezoid is approximately 3.999 cm.

Answer

The length of the smaller lateral side of the trapezoid is 2.308 cm. The length of the larger lateral side of the trapezoid is approximately 3.999 cm.