Question

Как найти стороны параллелограмма АBCD,если известно что его диагонали равны 10см и 16см,а угол между ними равен 60 градусов.Прошу с расширенным решением.

Answer 1

В параллелограмме ABCD, точка О пересечения диагоналей, из треугольника АОВ найдем $AB=a$ по теореме косинусов  
$a^2=5^2+8^2-2*5*8*cos60=25+64-80* \frac{1}{2}=49;a=7;$
Из треугольника ВОС найдем $BC=b$ по теореме косинусов  
$b^2=25+64+80*\frac{1}{2}=129;b= \sqrt{129}.$

Answer 2

Finding the Sides of a Parallelogram

To find the sides of a parallelogram ABCD when the diagonals are known to be 10 cm and 16 cm, and the angle between them is 60 degrees, we can use the following steps:

1. Use the Formula for Finding the Sides of a Parallelogram: - The sides of a parallelogram can be found using the formula: \[ \text{Side} = \sqrt{a^2 + b^2 + 2ab\cos(\theta)} \] where \(a\) and \(b\) are the lengths of the diagonals, and \(\theta\) is the angle between the diagonals.

2. Apply the Formula: - Given that the diagonals are 10 cm and 16 cm, and the angle between them is 60 degrees, we can substitute these values into the formula to find the sides of the parallelogram.

3. Calculate the Sides: - Using the given values, we can calculate the sides of the parallelogram using the formula.

4. Final Calculation: - After substituting the values into the formula, we can calculate the sides of the parallelogram.

Calculation

Using the formula for finding the sides of a parallelogram, we can calculate the sides of the parallelogram ABCD.

Let's denote: - \(a = 10\) cm (length of one diagonal) - \(b = 16\) cm (length of the other diagonal) - \(\theta = 60^\circ\) (angle between the diagonals)

Substituting these values into the formula: \[ \text{Side} = \sqrt{10^2 + 16^2 + 2(10)(16)\cos(60^\circ)} \]

\[ \text{Side} = \sqrt{100 + 256 + 320\cos(60^\circ)} \]

\[ \text{Side} = \sqrt{356 + 320 \times \frac{1}{2}} \]

\[ \text{Side} = \sqrt{356 + 160} \]

\[ \text{Side} = \sqrt{516} \]

\[ \text{Side} \approx 22.72 \, \text{cm} \]

So, the sides of the parallelogram ABCD are approximately 22.72 cm.

This calculation provides the approximate length of the sides of the parallelogram based on the given information about the diagonals and the angle between them.

Answer 3

Finding the Sides of a Parallelogram with Given Diagonals and Angle

To find the sides of a parallelogram ABCD, given that its diagonals are 10 cm and 16 cm long, and the angle between them is 60 degrees, we can use the properties of parallelograms and trigonometry.

Let's denote the length of one side of the parallelogram as a and the length of the adjacent side as b.

Using the Properties of Parallelograms

In a parallelogram, the diagonals bisect each other. This means that the length of each half of a diagonal is equal to half the length of the other diagonal. Therefore, we can find the lengths of the halves of the diagonals:

Half of the first diagonal = 10 cm / 2 = 5 cm

Half of the second diagonal = 16 cm / 2 = 8 cm

Now, we can use the properties of parallelograms to find the lengths of the sides.

Applying Trigonometry

Since we know the angle between the diagonals, we can use trigonometry to find the lengths of the sides.

In a parallelogram, the opposite sides are equal in length. Therefore, we have:

a = b

We can use the law of cosines to find the length of side a:

a^2 = (5 cm)^2 + (8 cm)^2 - 2 * (5 cm) * (8 cm) * cos(60 degrees)

Simplifying the equation:

a^2 = 25 cm^2 + 64 cm^2 - 80 cm^2 * cos(60 degrees)

a^2 = 89 cm^2 - 80 cm^2 * cos(60 degrees)

a^2 = 89 cm^2 - 80 cm^2 * 0.5

a^2 = 89 cm^2 - 40 cm^2

a^2 = 49 cm^2

Taking the square root of both sides:

a = sqrt(49 cm^2)

a = 7 cm

Therefore, the length of side a is 7 cm.

Since the opposite sides of a parallelogram are equal, the length of side b is also 7 cm.

Conclusion

The sides of the parallelogram ABCD, given that its diagonals are 10 cm and 16 cm long, and the angle between them is 60 degrees, are both 7 cm in length.

Please note that the solution provided is based on the given information and assumptions about the properties of parallelograms.