To find the value of [tex]\( p \)[/tex] such that [tex]\((-2, 2.5)\)[/tex] is the midpoint between [tex]\((p, 2)\)[/tex] and [tex]\((-1, 3)\)[/tex], follow these steps:
1. Understand the midpoint formula:
The midpoint [tex]\(M\)[/tex] of 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]
2. Assign the given values:
- The given midpoint is [tex]\((-2, 2.5)\)[/tex].
- One of the points is [tex]\((p, 2)\)[/tex].
- The other point is [tex]\((-1, 3)\)[/tex].
3. Set up equations using the midpoint formula:
Since [tex]\((-2, 2.5)\)[/tex] is the midpoint, we have:
[tex]\[
\left( \frac{p + (-1)}{2}, \frac{2 + 3}{2} \right) = (-2, 2.5)
\][/tex]
4. Separate into x and y components and solve each equation:
For the x-component, we have:
[tex]\[
\frac{p + (-1)}{2} = -2
\][/tex]
To solve for [tex]\( p \)[/tex], multiply both sides by 2:
[tex]\[
p - 1 = -4
\][/tex]
Then add 1 to both sides:
[tex]\[
p = -3
\][/tex]
For the y-component, we check:
[tex]\[
\frac{2 + 3}{2} = 2.5
\][/tex]
Simplified:
[tex]\[
\frac{5}{2} = 2.5
\][/tex]
This equation holds true, confirming our y-coordinate is correctly used.
5. Conclusion:
Hence, the value of [tex]\( p \)[/tex] that makes [tex]\((-2, 2.5)\)[/tex] the midpoint between [tex]\((p, 2)\)[/tex] and [tex]\((-1, 3)\)[/tex] is:
[tex]\[
p = -3
\][/tex]
