a certain city block is in the form of parallelogram two of its sides are each 35m long and the other side is 22m each if the distance between the first pair is 12 find the length of the diagonal

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!

A certain city block is in the form of parallelogram two of its sides are each 35m long and the other side is 22m each if the distance between the first pair is 12 find the length of the diagonal

Discover how to find the diagonal length of a parallelogram with sides measuring 35m and 22m, and a height of 12m. This solution employs trigonometric relationships and the Pythagorean theorem, suitable for Grades 9-12.

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