To determine the distance that a catcher throws the ball from home plate to second base in a square baseball diamond, we can use the properties of a square and the Pythagorean theorem. The baseball diamond forms a square with each side measuring the same length, which is 80 feet in this case.
1. Understanding the Square:
- Each side of the square baseball diamond is 80 feet.
- The throw from home plate to second base forms the diagonal of this square.
2. Using the Pythagorean Theorem:
- In a square, the diagonal splits the square into two right-angled triangles.
- For a right triangle with sides of equal length [tex]\(a\)[/tex], the length of the diagonal [tex]\(d\)[/tex] can be found using the Pythagorean theorem:
[tex]\[
d = \sqrt{a^2 + a^2}
\][/tex]
3. Applying the Side Length:
- Here, the side length [tex]\(a\)[/tex] is 80 feet, so we substitute [tex]\(a\)[/tex] into the formula:
[tex]\[
d = \sqrt{80^2 + 80^2}
\][/tex]
4. Simplifying the Expression:
- First, calculate [tex]\(80^2\)[/tex]:
[tex]\[
80^2 = 6400
\][/tex]
- Then,
[tex]\[
d = \sqrt{6400 + 6400} = \sqrt{12800}
\][/tex]
5. Simplifying the Square Root:
- The square root of 12800 can be simplified as follows:
[tex]\[
\sqrt{12800} = \sqrt{2 \times 6400} = \sqrt{2} \times \sqrt{6400} = \sqrt{2} \times 80
\][/tex]
- Therefore,
[tex]\[
d = 80\sqrt{2}
\][/tex]
The calculated distance from home plate to second base is [tex]\(80 \sqrt{2}\)[/tex] feet.
So, the correct answer is:
D. [tex]\( 80 \sqrt{2} \)[/tex] feet
