To find the value of [tex]\( p \)[/tex] that makes the points [tex]\((-2, -5)\)[/tex], [tex]\((2, -2)\)[/tex], and [tex]\((8, p)\)[/tex] collinear, we need to ensure that the slopes between each pair of points are equal. Here is the detailed step-by-step solution:
1. Calculate the slope between points [tex]\((-2, -5)\)[/tex] and [tex]\((2, -2)\)[/tex]:
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For points [tex]\((-2, -5)\)[/tex] and [tex]\((2, -2)\)[/tex]:
[tex]\[
\text{slope}_{AB} = \frac{-2 - (-5)}{2 - (-2)} = \frac{-2 + 5}{2 + 2} = \frac{3}{4} = 0.75
\][/tex]
2. Let the slope between points [tex]\((2, -2)\)[/tex] and [tex]\((8, p)\)[/tex] be equal to the slope calculated above, because the points need to be collinear:
For points [tex]\((2, -2)\)[/tex] and [tex]\((8, p)\)[/tex]:
[tex]\[
\text{slope}_{BC} = \frac{p - (-2)}{8 - 2} = \frac{p + 2}{6}
\][/tex]
3. Set the slope [tex]\(\text{slope}_{BC}\)[/tex] equal to the slope [tex]\(\text{slope}_{AB}\)[/tex] and solve for [tex]\(p\)[/tex]:
[tex]\[
\frac{p + 2}{6} = 0.75
\][/tex]
4. Solve the equation for [tex]\(p\)[/tex]:
[tex]\[
p + 2 = 0.75 \times 6
\][/tex]
[tex]\[
p + 2 = 4.5
\][/tex]
[tex]\[
p = 4.5 - 2
\][/tex]
[tex]\[
p = 2.5
\][/tex]
Thus, the value of [tex]\(p\)[/tex] that makes the points [tex]\((-2, -5)\)[/tex], [tex]\((2, -2)\)[/tex], and [tex]\((8, 2.5)\)[/tex] collinear is [tex]\(2.5\)[/tex].
Therefore, the correct value of [tex]\(p\)[/tex] is [tex]\(2.5\)[/tex].
