Question

1. a baseball diamond is a square that is 90 feet on each side. what is the distance catcher has to throw the ball from home to second base?

Explanation:

Step1: Identify the problem type

This is a right - triangle problem where the baseball diamond (a square) has sides of length \(a = b=90\) feet, and the distance from home to second base is the hypotenuse \(c\) of a right - triangle with legs equal to the side of the square. We can use the Pythagorean theorem, which states that for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\).

Step2: Substitute the values into the formula

Given \(a = 90\) and \(b = 90\), we substitute these values into the Pythagorean theorem:
\[

$$\begin{align*} c^{2}&=90^{2}+90^{2}\\ c^{2}&=8100 + 8100\\ c^{2}&=16200 \end{align*}$$

\]

Step3: Solve for \(c\)

To find \(c\), we take the square root of both sides:
\[
c=\sqrt{16200}=\sqrt{8100\times2}=90\sqrt{2}\approx90\times1.414 = 127.26
\]

Answer:

The distance from home to second base is \(90\sqrt{2}\approx127.27\) feet (or exactly \(90\sqrt{2}\) feet).