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.)
The distance from home plate to second base is
  
____________ feet

Answer 1

To find the distance from home plate to second base on a baseball diamond that is square-shaped, we can use the Pythagorean theorem.

In a square baseball diamond:
- Each side of the diamond (which is a square) measures 77 feet.
- Home plate and second base are two opposite corners of the square.

We can visualize this as a right triangle where:
- One leg (the distance from home plate to first base) is 77 feet.
- The other leg (the distance from first base to second base) is also 77 feet.

Using the Pythagorean theorem:

\[
d^2 = a^2 + b^2
\]

where \(d\) is the distance from home plate to second base, and \(a\) and \(b\) are the legs of the triangle.

Substituting the lengths:

\[
d^2 = 77^2 + 77^2
\]

Calculating \(77^2\):

\[
77^2 = 5929
\]

Then:

\[
d^2 = 5929 + 5929 = 11858
\]

Now, we can find \(d\):

\[
d = \sqrt{11858}
\]

To simplify \(\sqrt{11858}\):

\[
11858 = 2 \times 5929
\]

And since \(5929 = 77^2\):

\[
\sqrt{11858} = \sqrt{2 \times 77^2} = 77\sqrt{2}
\]

So, the exact distance from home plate to second base is:

\[
77\sqrt{2} \text{ feet}
\]

Now, for the decimal approximation:

\[
\sqrt{2} \approx 1.414
\]

Therefore:

\[
d \approx 77 \times 1.414 \approx 108.382
\]

Thus, we have:

- The distance from home plate to second base is \( 77\sqrt{2} \) feet (exact answer).
- The decimal approximation is approximately \( 108.38 \) feet.