To determine the location of \(\sqrt{78}\) on the number line, we first need to evaluate its approximate value.
The square roots of perfect squares close to 78 are:
- \(\sqrt{64} = 8\)
- \(\sqrt{81} = 9\)
Since \(78\) is between \(64\) and \(81\), it follows that:
\[ 8 < \sqrt{78} < 9 \]
To find a more precise estimate, we can calculate \(\sqrt{78}\).
Using a calculator, we find:
\[ \sqrt{78} \approx 8.83 \]
Now, placing \(8.83\) on the number line with the options provided:
- \(6\)
- \(7\)
- \(8\)
- \(9\)
- \(10\)
- \(11\)
- \(12\)
Since \(8.83\) falls between \(8\) and \(9\), the correct choice indicating the position of \(\sqrt{78}\) is:
[Choice C] \(8\)
So, the answer is Choice C, 8.