To find the length of the diagonal of a parallelogram, we can use the formula derived from the cosine rule. For a parallelogram with sides $a$ and $b$, and an angle $\theta$ between them, the lengths of the diagonals $d_1$ and $d_2$ can be calculated as follows:

$d_1 = \sqrt{a^2 + b^2 + 2ab \cos(\theta)}$ $d_2 = \sqrt{a^2 + b^2 - 2ab \cos(\theta)}$

In your case:

• $a = 35 \, m$
• $b = 22 \, m$
• The distance between the two sides (height) is given as 12 m, but we need the angle $\theta$ or the diagonal directly.

Instead, we can find the length of the diagonal using the relationship between the sides and the height. Since the height forms a right triangle with the sides of the parallelogram, we can use the Pythagorean theorem.

The formula for the diagonals can also be simplified for our situation. The diagonals of a parallelogram can be expressed as:

$d = \sqrt{a^2 + b^2 - 2ab \cos(\theta)}$

Given that we don’t have the angle directly, we can compute using the following steps.

1. Use the height to determine the angle between the sides.

The height $h$ and side $b$ (22 m) can help us find the angle:

$\sin(\theta) = \frac{h}{b} \implies h = b \sin(\theta) \implies 12 = 22 \sin(\theta)$

$\sin(\theta) = \frac{12}{22} = \frac{6}{11}$

Now, we can find $\cos(\theta)$ using $\sin^2(\theta) + \cos^2(\theta) = 1$:

$\cos^2(\theta) = 1 - \left(\frac{6}{11}\right)^2 = 1 - \frac{36}{121} = \frac{85}{121}$ $\cos(\theta) = \sqrt{\frac{85}{121}} = \frac{\sqrt{85}}{11}$

2. Substituting into the diagonal formula.

Now we can substitute back into the diagonal formula for one of the diagonals $d_1$:

$d = \sqrt{35^2 + 22^2 - 2 \cdot 35 \cdot 22 \cdot \frac{\sqrt{85}}{11}}$

Calculating this step by step:

$d = \sqrt{1225 + 484 - 2 \cdot 35 \cdot 22 \cdot \frac{\sqrt{85}}{11}}$ $d = \sqrt{1709 - \frac{1540\sqrt{85}}{11}}$

To simplify this further, let's calculate numerically:

$35^2 = 1225, \quad 22^2 = 484, \quad 2 \cdot 35 \cdot 22 = 1540$

Thus, we compute:

$d \approx \sqrt{1709 - 140.68} \text{ (where } \sqrt{85} \approx 9.22)$ $d \approx \sqrt{1568.32} \approx 39.6 \text{ m}$

Result

The length of the diagonal is approximately $39.6 \, m$.

Would you like more details on any of the calculations or concepts? Here are some related questions to consider:

1. How do you calculate the area of a parallelogram?
2. What properties do the diagonals of a parallelogram have?
3. How does the height relate to the angles in a parallelogram?
4. Can you derive the formula for the diagonal without using trigonometric functions?
5. What is the significance of the sides being equal in a parallelogram?

Tip: Always check if you have sufficient information before proceeding with formulas, especially in geometry!