Pythagorean theorem

[Dannielle.]

Location of a pitcher. A baseball diamond is officially defined as a square with side length of 90ft. The pitcher throws the ball to the batter from a position of 60ft. 6 in. away from home plate. Is the pitcher's position in front of, on, or behind the line from first base to third base? Use mathematics to justify answer.

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I understand that both the Pythagorean Theorem and formulas for special 45-45-90(degree) triangles can be used for this. I have concluded based on both formulas that the distance from first to 3rd base is 127.279ft. I'm not sure where to go from this point though.

 

[daon2]

Location of a pitcher. A baseball diamond is officially defined as a square with side length of 90ft. The pitcher throws the ball to the batter from a position of 60ft. 6 in. away from home plate. Is the pitcher's position in front of, on, or behind the line from first base to third base? Use mathematics to justify answer.

---

I understand that both the Pythagorean Theorem and formulas for special 45-45-90(degree) triangles can be used for this. I have concluded based on both formulas that the distance from first to 3rd base is 127.279ft. I'm not sure where to go from this point though.

You are to find the distance from home plate to the line passing through 1st and 3rd base. That is, if a batter were to walk directly from the home plate toward 2nd base, how far does the batter walk before he passes by 1st and third. We know that 2nd base lies at an angle of 45 degrees from 1st base relative to home plate.

The right triangle formed by the lines 1) from 1st to third and 2) from home to 2nd will give you your answer. The hypotenuse is 90 feet, so the distance along line 2) to line 1) is 90*cos(45 degrees)

 

[Dannielle.]

Pythagorean Theorem

Location of a pitcher. A baseball diamond is officially defined as a square with side length of 90ft. The pitcher throws the ball to the batter from a position of 60ft. 6 in. away from home plate. Is the pitcher's position in front of, on, or behind the line from first base to third base? Use mathematics to justify answer.

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Yesterday I posted this and was very confused with what I was doing. Today I took another look at it and realized that I was trying to find the length of the wrong line anyway. Now my work is :

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I drew a square and split it into four triangles, then used the Pythagorean theorem. I got x^2+x^2=90^2... 2x^2=8100... 8100/2=4050... Then the square root of 4050 being 63.63... Was this the correct process I was supposed to go through? And if so.. This means that the correct answer would be, the position of the pitcher is in front of the line from first to third base because according to the Pythagorean Theorem, the line from first to third base is 63.63 feet from home plate?

 

[srmichael]

Location of a pitcher. A baseball diamond is officially defined as a square with side length of 90ft. The pitcher throws the ball to the batter from a position of 60ft. 6 in. away from home plate. Is the pitcher's position in front of, on, or behind the line from first base to third base? Use mathematics to justify answer.

---

Yesterday I posted this and was very confused with what I was doing. Today I took another look at it and realized that I was trying to find the length of the wrong line anyway. Now my work is :

---

I drew a square and split it into four triangles, then used the Pythagorean theorem. I got x^2+x^2=90^2... 2x^2=8100... 8100/2=4050... Then the square root of 4050 being 63.63... Was this the correct process I was supposed to go through? And if so.. This means that the correct answer would be, the position of the pitcher is in front of the line from first to third base because according to the Pythagorean Theorem, the line from first to third base is 63.63 feet from home plate?

Correct. The pitcher is in front of the imaginary line connecting first base to third base. That line is just the diagnoal of the square (the baseball diamond). I think it is easier to relate where the pitcher is in relation to the diagonal from home plate to second base. This will also be $\displaystyle 90\sqrt{2} \approx 127.28$ ft and thus half way from home to second is about 63.64 ft (and lies on the diagonal from first to third) and since the distance from home plate to the pitcher is 60.5 ft, the pitcher is in front of the line from first to third base.