UpStudy Free Solution:

To solve the problem, let's analyze the given trapezoid in the diagram.

Part A: What is the length of the top?

From the diagram, we know that the height of the trapezoid is 9 miles, which is also the shortest side of the park. The top side of the trapezoid is the shortest side, so its length is 9 miles.

Answer to Part A: 9 miles

Part B: What is the length of the bottom?

The bottom side of the trapezoid is the sum of the top side and the two additional segments on either side. Each of these segments is 3 miles long.

So, the length of the bottom side is:

\[9 \text { miles ( top side) } + 3 \text { miles ( left segment) } + 3 \text { miles ( right segment) } = 15 \text { miles} \]

Answer to Part B: 15 miles

Part C: What is the area of the park?

The area \(A\) of a trapezoid is given by the formula:

\[A = \frac { 1} { 2} \times ( \text { Base} _ 1 + \text { Base} _ 2) \times \text { Height} \]

Where:

- \(\text { Base} _ 1\) is the length of the top side (9 miles)

- \(\text { Base} _ 2\) is the length of the bottom side (15 miles)

- \(\text { Height} \) is the height of the trapezoid (9 miles)

Plugging in the values:

\[A = \frac { 1} { 2} \times ( 9 \text { miles} + 15 \text { miles} ) \times 9 \text { miles} \]

\[A = \frac { 1} { 2} \times 24 \text { miles} \times 9 \text { miles} \]

\[A = \frac { 1} { 2} \times 216 \text { square miles} \]

\[A = 108 \text { square miles} \]

Answer to Part C: 108 square miles

Supplemental Knowledge

Understanding the properties of trapezoids and how to calculate their area is essential in geometry. Here’s a deeper dive into these concepts:

1. Trapezoid Definition: A trapezoid (or trapezium) is a quadrilateral with at least one pair of parallel sides. The parallel sides are called the bases, while the non-parallel sides are called the legs.

2. Key Properties:

- The height (or altitude) of a trapezoid is the perpendicular distance between its bases.

- The area of a trapezoid can be calculated using the formula:

\[A = \frac { 1} { 2} \times ( \text { Base} _ 1 + \text { Base} _ 2) \times \text { Height} \]

3. Example Calculation:

- Given:

- Top base (\(\text { Base} _ 1\)) = 9 miles

- Bottom base (\(\text { Base} _ 2\)) = 15 miles

- Height = 9 miles

- Plugging these values into the area formula:

\[A = \frac { 1} { 2} \times ( 9 + 15) \times 9\]

- Simplifying inside the parentheses first:

\[A = \frac { 1} { 2} \times 24 \times 9\]

- Multiplying:

\[A = 12 \times 9 = 108\]

Thus, the area of this trapezoid is \(108\) square miles.

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