To find the point that best represents the location for the science project to be equidistant from the two given points [tex]\( P \left(2, -15\right) \)[/tex] and [tex]\( Q \left(6.5, 6\right) \)[/tex], we can use the midpoint formula. The midpoint formula states that the midpoint [tex]\( M \)[/tex] between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Here, we are given:
- [tex]\( x_1 = 2 \)[/tex]
- [tex]\( y_1 = -15 \)[/tex]
- [tex]\( x_2 = 6.5 \)[/tex]
- [tex]\( y_2 = 6 \)[/tex]
Let's calculate the coordinates of the midpoint step-by-step.
1. Find the midpoint of the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Midpoint}_x = \frac{x_1 + x_2}{2} = \frac{2 + 6.5}{2} = \frac{8.5}{2} = 4.25 \][/tex]
2. Find the midpoint of the [tex]\( y \)[/tex]-coordinates:
[tex]\[ \text{Midpoint}_y = \frac{y_1 + y_2}{2} = \frac{-15 + 6}{2} = \frac{-9}{2} = -4.5 \][/tex]
Therefore, the coordinates of the midpoint [tex]\( M \)[/tex] are [tex]\( (4.25, -4.5) \)[/tex].
Thus, the point that best represents the location for them to put the science project, so that both observers are the same distance from it, is at [tex]\( (4.25, -4.5) \)[/tex].
