Let's go through the problem step by step.
Given:
- The coordinates of Point P are [tex]\((6, 6)\)[/tex].
- The coordinates of Point Q are [tex]\((7, 7)\)[/tex].
We need to find the midpoint between these two points so that it best represents the location for the science project. The midpoint formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Substitute the coordinates of Point P and Point Q into the formula:
[tex]\[
x_1 = 6, \quad y_1 = 6, \quad x_2 = 7, \quad y_2 = 7
\][/tex]
Now, calculate the midpoint:
[tex]\[
\left( \frac{6 + 7}{2}, \frac{6 + 7}{2} \right) = \left( \frac{13}{2}, \frac{13}{2} \right) = (6.5, 6.5)
\][/tex]
Thus, the point that best represents the location for the science project, so that both observers are the same distance from it, is [tex]\((6.5, 6.5)\)[/tex].
Now, let's compare this result with the given options:
a. [tex]\((6.5, 6)\)[/tex]
b. [tex]\((-15, 2)\)[/tex]
c. [tex]\((6, 6.5)\)[/tex]
None of these match our calculated midpoint of [tex]\((6.5, 6.5)\)[/tex]. It seems the exact calculated midpoint is not listed as an option. If we had to choose a point among the given options, [tex]\((6.5, 6)\)[/tex] and [tex]\((6, 6.5)\)[/tex] are closer to the ideal midpoint than [tex]\((-15, 2)\)[/tex].
However, according to our exact theoretical calculations, the correct answer should be [tex]\((6.5, 6.5)\)[/tex].
