Question

Сторона прямоугольника равна 8 см и образует с диагональю угол 30 градусов найдите площадь прямоугольника

Answer 1

Диагональ поделила треугольник на 2 треугольника
Рассмотрим треугольник:
Мы знаем теорему синусов
Катет лежащий напротив угла в 30 градусов=половине гипотенузы
Гипотенуза равна 8
Значит катет равен 4(который лежит напротив угла в 30 гр)
Площадь треугольника равна умножить сторону на сторону(а*б)
Тогда s=4*8=32

Answer 2

Problem Analysis

We are given that one side of a rectangle is 8 cm and it forms a 30-degree angle with the diagonal. We need to find the area of the rectangle.

Solution

To find the area of the rectangle, we need to know the length of the other side. Let's call the length of the other side "x".

We can use trigonometry to find the length of the diagonal. The diagonal is the hypotenuse of a right triangle formed by one side of the rectangle and the other side.

Using the given angle of 30 degrees and the length of one side (8 cm), we can use the sine function to find the length of the diagonal:

sin(30) = opposite/hypotenuse

Since the opposite side is 8 cm (one side of the rectangle) and we want to find the hypotenuse (diagonal), we can rearrange the equation:

hypotenuse = opposite / sin(30)

Now we can substitute the values and calculate the length of the diagonal:

hypotenuse = 8 cm / sin(30 degrees)

Let's calculate the value of sin(30 degrees) using a calculator:

sin(30 degrees) ≈ 0.5

Substituting this value into the equation:

hypotenuse = 8 cm / 0.5 ≈ 16 cm

So, the length of the diagonal is approximately 16 cm.

Now that we know the length of the diagonal and one side of the rectangle, we can use the formula for the area of a rectangle:

Area = length × width

Substituting the values:

Area = 8 cm × 16 cm = 128 cm²

Therefore, the area of the rectangle is 128 square centimeters.

Answer

The area of the rectangle is 128 square centimeters.