In a baseball field, home plate and the three bases are supposed to be the corners of a square. Do the directions above result in home plate and the three bases being the corners of a square?

Question

1. Give a description of how you would use the Pythagorean Theorem to determine whether creating home plate and the 3 bases as directed above creates a square. If it doesn't make a square, how close is it to being a square (your answer should be a number, not words like "very close")?
2. Give a description of how you would use the Law of Cosines to determine whether creating home plate and the 3 bases as directed above creates a right angle at first base. If it doesn't make a right angle at first base, how close is it to being a right angle (Again, your answer should be a number, not words like "really close")?

Please help!!!

Answer 1

The description of this problem is a little bit weird. It feels like something was left out. If you say home plate and the 3 bases are supposed to be the corners of a square, then by definition home plate and the 3 bases should be corners of a square. QED.

Having said that, here are some details that should help you answer the question:

If you divide a square into two triangles, you will get two isosceles right triangles with legs the length of the sides of the square. Angles will be 45/45/90.

You can do this with the bases themselves or with the baselines around the infield. You can measure the hypotenuse in either case.

Pythagorean theorem: If the sides of the triangle are equal, then x^2 + y^2 = h^2 can be simplified to 2x^2 = h^2. Substitute the length of one leg and the hypotenuse and see if the result is correct.

Cosine law: If the sides of the triangle are equal, then a and b are the same value and theta is 90 degrees. This will allow you to dramatically simplify the equation, and then verify with actual values.