Question

Find the third side in simplest radical form:
Side A - Square root 97
Side b - 1
Side C - ?

Answer 1

To find the third side \( C \) of a triangle when given two sides \( A \) and \( B \), we can use the triangle inequality theorem, which states:

1. \( A + B > C \)
2. \( A + C > B \)
3. \( B + C > A \)

Given:
- \( A = \sqrt{97} \)
- \( B = 1 \)

Let's denote the third side as \( C \). We will work through the inequalities.

### Step 1: Use the triangle inequalities

1. From \( A + B > C \):
\[
\sqrt{97} + 1 > C
\]
This means \( C < \sqrt{97} + 1 \).

2. From \( A + C > B \):
\[
\sqrt{97} + C > 1
\]
This gives \( C > 1 - \sqrt{97} \). Given that \( \sqrt{97} \) is larger than 1, this inequality does not provide a useful lower bound.

3. From \( B + C > A \):
\[
1 + C > \sqrt{97}
\]
Rearranging gives:
\[
C > \sqrt{97} - 1
\]

### Step 2: Combine the inequalities

Now we can summarize the inequalities for \( C \):
\[
\sqrt{97} - 1 < C < \sqrt{97} + 1
\]

### Step 3: Find a specific value for \( C \)

Since we don't know if \( C \) is at its maximum or minimum value from the inequalities, we may need more information about the kind of triangle (e.g., acute, right, or obtuse). However, if we were to find \( C \) such that it is directly related (using a Pythagorean theorem case), we can assume it might have a specific value.

If it's a right triangle, then:
\[
C^2 = A^2 + B^2
\]
Calculating:
\[
A^2 = 97
\]
\[
B^2 = 1
\]
Adding these:
\[
C^2 = 97 + 1 = 98
\]
So, we have:
\[
C = \sqrt{98}
\]
This can be simplified:
\[
C = \sqrt{49 \times 2} = 7\sqrt{2}
\]

### Conclusion

The third side \( C \) in simplest radical form, assuming a right triangle scenario, is:
\[
\boxed{7\sqrt{2}}
\]