Question

Діагональ прямокутника дорівнює 12 см і утворює з його стороною кут 60 градусів. знайдіть сторони прямокутника.

Answer 1

Діагональ утворює зі стронами трикутника прямокутний трикутник.Тоді
за т.Піфагора
a=d·cosa ,b=d·sina
a=12·cos60°=12·0/5=6
b=12·sin60°=12·√3/2=6√3

Answer 2

Problem Analysis

We are given that the diagonal of a rectangle measures 12 cm and forms a 60-degree angle with one of its sides. We need to find the lengths of the sides of the rectangle.

Solution

Let's assume the length of the rectangle is L and the width is W. We can use the given information to set up the following equations:

1. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the lengths of the other two sides. In this case, we have a right triangle formed by the diagonal and one side of the rectangle. Therefore, we can write the equation as:

L^2 + W^2 = 12^2 2. We are also given that the diagonal forms a 60-degree angle with one of the sides. In a rectangle, opposite sides are equal in length. Therefore, the side of the rectangle that forms a 60-degree angle with the diagonal is equal to the width of the rectangle. We can use trigonometry to find the relationship between the width and the diagonal. In a right triangle, the sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. In this case, we have:

sin(60) = W / 12

Simplifying this equation, we get:

W = 12 * sin(60) Now, let's solve these equations to find the lengths of the sides of the rectangle.

Calculation

Using equation we can calculate the width of the rectangle:

W = 12 * sin(60) = 12 * 0.866 = 10.392 cm

Using equation we can substitute the value of W and solve for L:

L^2 + (10.392)^2 = 12^2

L^2 + 107.67 = 144

L^2 = 144 - 107.67

L^2 = 36.33

L = sqrt(36.33) = 6.03 cm

Therefore, the length of the rectangle is approximately 6.03 cm and the width is approximately 10.392 cm.

Answer

The sides of the rectangle are approximately 6.03 cm and 10.392 cm.

Note: The values are approximate due to rounding.