Question

Стороны параллелограмма равны 6 и 4. Угол между его диагоналями равен 45. Найти площадь.

Answer 1

Given Information

We are given that the sides of the parallelogram are equal to 6 and 4, and the angle between its diagonals is 45 degrees. We need to find the area of the parallelogram.

Solution

To find the area of the parallelogram, we can use the formula: Area = base * height.

Finding the Base

The base of the parallelogram is one of its sides. We are given that the sides of the parallelogram are equal to 6 and 4. Since opposite sides of a parallelogram are equal, we can choose either side as the base. Let's choose the side of length 6 as the base.

Finding the Height

To find the height of the parallelogram, we need to find the length of the perpendicular dropped from one of the vertices to the base. Since we don't have the height directly given, we can use trigonometry to find it.

Using Trigonometry

We are given that the angle between the diagonals is 45 degrees. Let's assume that the diagonals of the parallelogram intersect at point O. We can draw a right triangle OAB, where OA is the base of the parallelogram and AB is the height.

Using trigonometry, we can find the length of AB. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the opposite side is AB and the adjacent side is OA.

The tangent of 45 degrees is 1. So we have:

tan(45) = AB / OA

Since AB is the height of the parallelogram and OA is the base, we can rewrite the equation as:

1 = AB / 6

Simplifying the equation, we find:

AB = 6