Question

Диагонали ромба равны 10 см и 10 корней из 3 см .найдите углы ромба .решение и дано

Answer 1

Дано ромб ABCD
AC=10 корень из 3 см
BD=10 см
О-точка пересечения диагоналей
Найти
углы ромба
Решение
диагонали ромба взаимноперпендикулярны и точкой пересечения делятся пополам => AO=ОС=10 корень из 3 / 2 = 5 корень из 3 см
BO=OD=10/2=5 см
рассм. ABO: tg B=AO/BO=5корень из 3 / 5 = корень из 3 => угол B=60 гр
диагонали ромба делят углы пополам => угол B=60*2=120
углы ромба попарно равны => угол B=углу D=120, угол A=углу С=(360-120-120)/2=60

Answer 2

Finding the Angles of a Rhombus

To find the angles of a rhombus when the diagonals are given, we can use the formula:

\( \cos(\theta) = \frac{d_1^2 + d_2^2 - s^2}{2 \cdot d_1 \cdot d_2} \)

Where: - \( \theta \) = the angle between the diagonals - \( d_1 \) = length of one diagonal - \( d_2 \) = length of the other diagonal - \( s \) = side length of the rhombus

Given: - \( d_1 = 10 \, \text{cm} \) - \( d_2 = 10\sqrt{3} \, \text{cm} \)

Let's calculate the angle \( \theta \) using the given formula.

Using the given values: - \( d_1 = 10 \, \text{cm} \) - \( d_2 = 10\sqrt{3} \, \text{cm} \)

We can calculate the angle \( \theta \) as follows:

\( \cos(\theta) = \frac{10^2 + (10\sqrt{3})^2 - s^2}{2 \cdot 10 \cdot 10\sqrt{3}} \)

\( \cos(\theta) = \frac{100 + 300 - s^2}{20\sqrt{3}} \)

\( \cos(\theta) = \frac{400 - s^2}{20\sqrt{3}} \)

Now, we need the value of \( s \) to calculate the angle \( \theta \).

Calculating the Side Length of the Rhombus

The side length \( s \) of a rhombus can be calculated using the formula:

\( s = \sqrt{\frac{d_1^2 + d_2^2}{2}} \)

Given: - \( d_1 = 10 \, \text{cm} \) - \( d_2 = 10\sqrt{3} \, \text{cm} \)

We can calculate the side length \( s \) as follows:

\( s = \sqrt{\frac{10^2 + (10\sqrt{3})^2}{2}} \) \( s = \sqrt{\frac{100 + 300}{2}} \) \( s = \sqrt{\frac{400}{2}} \) \( s = \sqrt{200} \) \( s = 10\sqrt{2} \, \text{cm} \)

Now that we have the value of \( s \), we can substitute it into the equation for \( \cos(\theta) \) to find the angle \( \theta \).

Calculating the Angle \( \theta \)

Substituting the value of \( s = 10\sqrt{2} \, \text{cm} \) into the equation for \( \cos(\theta) \):

\( \cos(\theta) = \frac{400 - (10\sqrt{2})^2}{20\sqrt{3}} \) \( \cos(\theta) = \frac{400 - 200}{20\sqrt{3}} \) \( \cos(\theta) = \frac{200}{20\sqrt{3}} \) \( \cos(\theta) = \frac{10}{\sqrt{3}} \) \( \theta = \arccos\left(\frac{10}{\sqrt{3}}\right) \) \( \theta \approx 19.471 \, \text{degrees} \)

So, the angle \( \theta \) between the diagonals of the rhombus is approximately 19.471 degrees.