Let's solve the problem step-by-step.
### Part A: Calculating the Slope from [tex]\((-8, 9)\)[/tex] to [tex]\((-4, 6)\)[/tex]
To find the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], we use the slope formula:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For points [tex]\((-8, 9)\)[/tex] and [tex]\((-4, 6)\)[/tex]:
[tex]\[
x_1 = -8, \quad y_1 = 9, \quad x_2 = -4, \quad y_2 = 6
\][/tex]
Plug these values into the formula:
[tex]\[
\text{slope}_a = \frac{6 - 9}{-4 - (-8)} = \frac{6 - 9}{-4 + 8} = \frac{-3}{4} = -0.75
\][/tex]
So, the slope from [tex]\((-8, 9)\)[/tex] to [tex]\((-4, 6)\)[/tex] is [tex]\(\mathbf{-0.75}\)[/tex].
### Part B: Calculating the Slope from [tex]\((-4, 6)\)[/tex] to [tex]\(\left(2, \frac{3}{2}\right)\)[/tex]
Using the same slope formula for points [tex]\((-4, 6)\)[/tex] and [tex]\(\left(2, \frac{3}{2}\right)\)[/tex]:
[tex]\[
x_1 = -4, \quad y_1 = 6, \quad x_2 = 2, \quad y_2 = \frac{3}{2}
\][/tex]
Plug these values into the formula:
[tex]\[
\text{slope}_b = \frac{\frac{3}{2} - 6}{2 - (-4)} = \frac{\frac{3}{2} - 6}{2 + 4} = \frac{\frac{3}{2} - \frac{12}{2}}{6} = \frac{\frac{3 - 12}{2}}{6} = \frac{\frac{-9}{2}}{6} = \frac{-9}{12} = \frac{-3}{4} = -0.75
\][/tex]
So, the slope from [tex]\((-4, 6)\)[/tex] to [tex]\(\left(2, \frac{3}{2}\right)\)[/tex] is [tex]\(\mathbf{-0.75}\)[/tex].
### Part C: Relationship Between the Points
We have calculated the slopes from the previous segments:
- Slope from [tex]\((-8, 9)\)[/tex] to [tex]\((-4, 6)\)[/tex] is [tex]\(-0.75\)[/tex]
- Slope from [tex]\((-4, 6)\)[/tex] to [tex]\(\left(2, \frac{3}{2}\right)\)[/tex] is [tex]\(-0.75\)[/tex]
Since both slopes are equal ([tex]\(-0.75\)[/tex]), all three points lie on the same straight line. This indicates that the points [tex]\((-8, 9)\)[/tex], [tex]\((-4, 6)\)[/tex], and [tex]\(\left(2, \frac{3}{2}\right)\)[/tex] are collinear.
Therefore, the relationship between the points is that they all lie on the same straight line.
