Question

Диагональ прямоугольного треугольника равна 12 см с одной из его сторон угол 30. найти площадь

Answer 1

Известно, что катет прямоугольного треугольника, лежащий против угла в 30 градусов, равен половине гипотенузы. Значит, один из катетов треугольника будет равен 12/2=6. Зная величину гипотенузы и одного из катетов, найдём величину второго катета по теореме Пифагора - он будет равен √12²-6²=√144-36=√108=6√3. Таким образом, катеты нашего треугольника равны 6 и 6√3. Площадь прямоугольного треугольника равна половине произведения катетов, то есть, S=1/2*6*6√3=18√3см².

Answer 2

Given Information

We are given that the diagonal of a right triangle is 12 cm and one of its sides forms a 30-degree angle.

Solution

To find the area of the triangle, we need to know the lengths of the two sides that form the right angle. Let's denote these sides as a and b.

Using trigonometric ratios, we can relate the lengths of the sides to the given angle. In this case, we know that one of the angles is 30 degrees. The trigonometric ratio for the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. The trigonometric ratio for the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

In our case, the side opposite the 30-degree angle is a and the hypotenuse is 12 cm. Therefore, we can write:

sin(30 degrees) = a / 12

Similarly, the side adjacent to the 30-degree angle is b. Therefore, we can write:

cos(30 degrees) = b / 12

Simplifying these equations, we have:

a = 12 * sin(30 degrees)

b = 12 * cos(30 degrees)

Now that we have the lengths of the two sides, we can calculate the area of the triangle using the formula:

Area = (1/2) * base * height

In this case, the base is a and the height is b. Therefore, the area of the triangle is:

Area = (1/2) * a * b

Substituting the values of a and b we obtained earlier, we have:

Area = (1/2) * (12 * sin(30 degrees)) * (12 * cos(30 degrees))

Simplifying further, we get:

Area = 36 * sin(30 degrees) * cos(30 degrees)

Using the trigonometric identity sin(2θ) = 2 * sin(θ) * cos(θ), we can simplify the expression:

Area = 36 * (1/2) * sin(60 degrees)

Since sin(60 degrees) = √3/2, we can substitute this value:

Area = 36 * (1/2) * (√3/2)

Finally, calculating the value, we get:

Area = 18 * √3 square cm.

Therefore, the area of the given right triangle is 18√3 square cm.