Question

A particular baseball diamond is actually a square with 77​-foot sides. What is the distance from home plate to second​ base? Express the answer in simplified radical form. Then find a decimal approximation.
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Part 1
The distance from home plate to second base is
  
enter your response here feet.
​(Type an exact​ answer, using radicals as​ needed.)

Answer 1

To find the distance from home plate to second base in a baseball diamond, which is shaped like a square, we can use the Pythagorean theorem. In this case, home plate and second base are opposite corners of the square.

Let the sides of the square be \( s = 77 \) feet. In a right triangle formed by home plate, first base, and the line connecting home plate to second base, the two legs of the triangle are each \( 77 \) feet long.

According to the Pythagorean theorem:
\[
d = \sqrt{s^2 + s^2}
\]
\[
d = \sqrt{77^2 + 77^2}
\]
\[
d = \sqrt{2 \times 77^2}
\]
\[
d = \sqrt{2} \times 77
\]

Now let's simplify:
\[
d = 77\sqrt{2}
\]

This is the exact distance from home plate to second base in simplified radical form.

Now, for the decimal approximation:
\[
\sqrt{2} \approx 1.414
\]
So:
\[
d \approx 77 \times 1.414 \approx 108.569
\]

To summarize:

- The distance from home plate to second base is \( 77\sqrt{2} \) feet (exact).
- The decimal approximation is approximately \( 108.57 \) feet.