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

Question

The Major League Baseball website provides instructions for laying out a basball field: “when the location of home plate is determined, use a steel tape (tape measure) to measure 127 feet, 3 3/8 inches in the desired direction to establish second base. From home plate measure 90 feet towards first base. From second base measure 90 feet towards first base. The intersection of these lines locates first base.” Third base is found in a similar manner, but to the left of home plate instead of to the right.

2. Give a description of how you would use the Pythagorean Theorem to determnine 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")?

3. 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 ASAP!!

 

Answer 1

home plate to 2nd base is the hypotenuse of a right triangle with 2 sides = 90 feet

h^2 = 90^2 + 90^2 = 2(90^2) = 2(8100) = 16200

h = sqr16200 = 127.2792206... = about 127.28 feet

127 feet and 3 3/8 inches = 127 + 0.28125 = 127.28125 which is virtually the same as 127.28

just off by 127.28125 - 127.28 = 0.00125 feet or 12(.00125 = 0.015 inches off

the angle at home place is slightly less than 90 degrees

the angle from line through the pitchers' mound to 1st or 3rd base is the sin^-1(90/127.28125) = 44.99908648

double it to get 89.99817295 which is

within ,0018270456 degrees of a right angle

the baseball diamond is a square, or virtually, almost, down to a couple decimal places.

Law of Cosines is a generalized version of the Pythagorean Theorem which is c^2= a^2+b^2.

for a right triangle with C =90, Cos90=0

the last term drops out.

c^2 = a^2 + b^2 -2abCosC

127.28125^2 = 90^2 + 90^2 -2(90)(90)CosC

--16200CosC = 16200.5166 - 16200 = .5166

CosC = ,5166/16200 = .00003188898148

C =89.9981729 = about 89.998 degrees<90

that's the angle at home plate formed by base lines to 1st and 3rd

instead of a perfect square, it's a parallelogram with opposite angles equal

the angles at 1st and 3rd bases are 180-89.998 = 90.002 degrees

the angle at 2nd base is the same as at home plate, 89.998 degrees