Question

В параллелограмме ABCD диагональ BD перпендикулярна стороне AB, один из углов параллелограмма равен 120 градусов,AD =12 см.,О- точка пересечения диагоналей. Найдите диагонали параллелограмма. Помогите пожалуйста !!

Answer 1

Угол СВД = углу ВДА = 30градусов
cosВДА=cos30=ВД/АД
$BD=cos30*12=6 \sqrt{3}$см

$AB=sin30*12=6$см

по теореме косинусов
$AC= \sqrt{AB^{2}+AD^{2}-2AD*AB*cos120} =$$\sqrt{6^{2}+12^{2}+2*6*12* \frac{1}{2} } =\sqrt{36+144+72} =6 \sqrt{7}$см

Answer 2

Given Information

We are given the following information about the parallelogram ABCD: - Diagonal BD is perpendicular to side AB. - One of the angles of the parallelogram is 120 degrees. - AD = 12 cm. - O is the point of intersection of the diagonals.

Finding the Diagonals of the Parallelogram

To find the diagonals of the parallelogram, we can use the properties of a parallelogram.

Property 1: In a parallelogram, opposite sides are equal in length. Property 2: In a parallelogram, opposite angles are equal in measure.

Since BD is perpendicular to AB, we can conclude that the parallelogram is a rectangle. In a rectangle, all angles are 90 degrees.

Let's denote the length of the diagonals as AC and BD.

From Property 1, we know that AC = BD.

From Property 2, we know that angle ABD = angle BAC = 90 degrees.

Since one of the angles of the parallelogram is 120 degrees, we can conclude that angle ABC = 180 - 120 = 60 degrees.

Now, we have a right triangle ABD with angle ABD = 90 degrees and angle ABC = 60 degrees.

Using trigonometric ratios, we can find the length of AB.

Let's denote the length of AB as x.

In right triangle ABD, we have: - angle ABD = 90 degrees - angle ABC = 60 degrees - AD = 12 cm

Using the sine function, we can write: sin(60) = AB/AD

Simplifying the equation, we get: AB = AD * sin(60)

Substituting the given value, we have: AB = 12 cm * sin(60)

Calculating the value, we find: AB ≈ 12 cm * 0.866 ≈ 10.392 cm

Since opposite sides of a rectangle are equal, we can conclude that AC = BD = AB.

Therefore, the diagonals of the parallelogram are approximately 10.392 cm each.

Answer

The diagonals of the parallelogram ABCD are approximately 10.392 cm each.