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The problem describes a parallelogram-shaped city block with:

- Two parallel sides of 120 meters,
- The other two sides of 76 meters,
- The height (distance between the first pair of parallel sides) is 50 meters.

The goal is to find the two diagonals of the parallelogram.

To solve this, we will use the following formulae for the diagonals of a parallelogram:

1. **Diagonal 1** \((d_1)\):
\[
d_1 = \sqrt{a^2 + b^2 + 2ab \cos \theta}
\]
2. **Diagonal 2** \((d_2)\):
\[
d_2 = \sqrt{a^2 + b^2 - 2ab \cos \theta}
\]

Where:
- \(a = 120\) meters (one pair of sides),
- \(b = 76\) meters (other pair of sides),
- \(\theta\) is the angle between the sides, which can be found using the formula:
\[
\sin \theta = \frac{\text{height}}{\text{side length}} = \frac{50}{120} = \frac{5}{12}
\]
Thus, \(\theta = \arcsin\left(\frac{5}{12}\right)\).

Let's calculate both diagonals.The two diagonals of the city block are approximately:

1. Diagonal 1: \(191.72\) meters
2. Diagonal 2: \(59.96\) meters

Would you like more details or explanations on how these values were calculated?

Here are 5 related questions to expand on this concept:

1. How does the area of a parallelogram relate to its diagonals?
2. What happens to the diagonals if the parallelogram becomes a rectangle?
3. How do you calculate the perimeter of the parallelogram?
4. What other ways can you find the angle between the sides of a parallelogram?
5. How would this change if the parallelogram was not symmetric?

**Tip:** In any parallelogram, the diagonals bisect each other but are not necessarily equal in length, unlike in a rectangle or square.

A certain city block is in the form of a parallelogram. Two of its parallel sides are each 120 meters long; the other two sides are each 76 meters in length. If the distance between the first two parallel sides is 50 meters, find the two diagonals of the city block.

Solve for the two diagonals of a parallelogram-shaped city block, where two sides are 120 meters, the other two sides are 76 meters, and the height is 50 meters. Using trigonometry and geometry, this problem is ideal for high school students in Grades 10-12.

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