Question

1. the figure shows a coordinate plane on a baseball field. the distance from home plate to first base is 90 feet. the pitching mound is the mid - point between home plate and second base. find the distance from home plate to second base. find the distance between home plate and the pitching mound. explain how you found your answers. (section 1.3)

28 chapter 1 basics of geometry

Explanation:

Step1: Recognize baseball - field shape

A baseball field is a square. The distance from home - plate to first - base is 90 feet, so the side length of the square field \(a = 90\) feet.

Step2: Use the Pythagorean theorem to find distance to second - base

In a right - triangle formed by two sides of the square (home - plate to first - base and first - base to second - base), if the side length of the square is \(a\), the distance \(d\) from home - plate to second - base is the hypotenuse of the right - triangle. By the Pythagorean theorem \(d=\sqrt{a^{2}+a^{2}}\). Substituting \(a = 90\) feet, we have \(d=\sqrt{90^{2}+90^{2}}=\sqrt{2\times90^{2}}=90\sqrt{2}\approx90\times1.414 = 127.26\) feet.

Step3: Find distance to pitching mound

The pitching mound is the mid - point between home - plate and second - base. So the distance \(m\) from home - plate to the pitching mound is half of the distance from home - plate to second - base. \(m=\frac{90\sqrt{2}}{2}=45\sqrt{2}\approx45\times1.414 = 63.63\) feet.

Answer:

The distance from home plate to second base is \(90\sqrt{2}\approx127.26\) feet. The distance from home plate to the pitching mound is \(45\sqrt{2}\approx63.63\) feet.