Question

Giselle is building a small baseball infield for her brother. The infield should be in the shape of a square. Each side of the infield, between bases, measures 30 feet. The distance between home plate and second base, as shown in the picture, is 45 feet.
A square has points home plate, first base, second base, third base. The distance between home plate and second base is 45 feet. The distance between home plate and first base is 30 feet.
Which best describes the shape of the baseball field?
Square: 30 squared + 30 squared = 45 squared
Not square: (30 + 30) squared not-equals 45 squared
Not square: 2 (30) squared not-equals 45 squared
Not square: (2 times 30) squared not-equals 45 squared

Answer 1

To determine whether the described baseball infield is a square, we can utilize the Pythagorean theorem. The distances between home plate and the bases can be visualized in a right triangle formed with:

- One leg as the distance from home plate to first base (30 feet).
- The other leg as the distance from first base to second base (also 30 feet, since it’s a square).
- The hypotenuse as the distance from home plate to second base (given as 45 feet).

Let's denote:
- a = 30 feet (home plate to first base)
- b = 30 feet (first base to second base)
- c = 45 feet (home plate to second base)

According to the Pythagorean theorem, for a right triangle:

\[ a^2 + b^2 = c^2 \]

Plugging in the known values:

\[ 30^2 + 30^2 = 45^2 \]
\[ 900 + 900 = 2025 \]
\[ 1800 \neq 2025 \]

Since \( 1800 \) does not equal \( 2025 \), the relationship does not hold.

Among the given options, the one that best describes the situation would be:

**Not square: (30 + 30) squared not-equals 45 squared**

This suggests the sum of the legs of the triangle leads us to the conclusion that the distances do not form a valid square shape for the baseball infield.