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The formula for the midpoint between two points [tex]\(\left( x_1, y_1 \right) \)[/tex] and [tex]\(\left( x_2, y_2 \right) \)[/tex] is [tex]\(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)[/tex].

Points [tex]\(P\)[/tex] and [tex]\(Q\)[/tex] on the coordinate grid below represent the locations of two observers for a science project.

If the two observers both want to be the same distance from the science project, and as close as possible to it, what point best represents the location for them to put the science project?

A. [tex]\((6.5, 6)\)[/tex]

B. [tex]\((-15, 2)\)[/tex]

C. [tex]\((6, 6.5)\)[/tex]


Sagot :

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].