To find out how far the shortstop needs to throw the ball to first base, we can use the Pythagorean theorem.
The baseball diamond is a square, and the distance between bases is 90 feet. Let's outline the positions:
- The distance from the shortstop to second base is 25 feet.
- Therefore, the distance from the shortstop to third base is \(90 - 25 = 65\) feet.
- The distance from second base to first base is 90 feet.
Now, when the shortstop fields the ball, we can consider the position of the shortstop as a point \(S\), second base as point \(B2\), third base as point \(B3\), and first base as point \(B1\). The coordinates of these points can be considered as:
- Second base (B2) at (0, 0)
- First base (B1) at (90, 0)
- Third base (B3) at (0, 90)
- The shortstop (S), who is 25 feet from B2, would be located at (0, -25).
Now, we need to find the distance from the shortstop \(S(0, -25)\) to first base \(B1(90, 0)\). We can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting in our coordinates:
\[ d = \sqrt{(90 - 0)^2 + (0 - (-25))^2} \]
This simplifies to:
\[ d = \sqrt{90^2 + 25^2} \] \[ d = \sqrt{8100 + 625} \] \[ d = \sqrt{8725} \]
Calculating this gives:
\[ d \approx 93.3 \text{ feet} \]
Therefore, the shortstop needs to throw the ball approximately 93.3 feet to first base.