Let's break down and solve each part of the question based on the image provided.

9.1. Name the quadrilateral formed by these bases.

The bases form a square. Since all sides are equal (27.4 m each), the quadrilateral formed is a square.

9.2. How far is the distance from home base to the second base if you move directly between the two bases?

The distance between the home base and the second base is the diagonal of the square. To find the diagonal $d$ of a square with side length $s$, we use the Pythagorean theorem: $d = s\sqrt{2}$ Given that $s = 27.4 \, \text{m}$, the diagonal is: $d = 27.4 \times \sqrt{2} \approx 27.4 \times 1.414 \approx 38.73 \, \text{m}$ So, the distance from the home base to the second base is approximately 38.73 meters.

9.3. Find the perimeter around the bases of the baseball field.

Since the quadrilateral is a square, the perimeter $P$ is given by: $P = 4 \times \text{side length}$ $P = 4 \times 27.4 \, \text{m} = 109.6 \, \text{m}$ Thus, the perimeter around the bases is 109.6 meters.

9.4. Determine the area covered by the bases.

The area $A$ of a square is given by: $A = \text{side length}^2$ $A = 27.4^2 = 750.76 \, \text{m}^2$ So, the area covered by the bases is 750.76 square meters.

Let me know if you'd like further details on any step or if you have any questions!

Additional Questions:

1. What other quadrilaterals have diagonals that use the Pythagorean theorem for diagonal calculation?
2. How can you calculate the diagonal of a rectangle if its sides are different lengths?
3. In what other sports might similar geometry (square layout) be found?
4. How could you calculate the diagonal of a regular pentagon?
5. How would the calculations change if the bases formed a rectangle instead of a square?

Tip:

Whenever you're asked to find the diagonal of a square, remember it’s always $\sqrt{2}$ times the side length. This trick saves you time!