Answer:
1. Slope: [tex]m = \frac{4}{3}[/tex]
2. Distance: [tex]AB = 10[/tex]
3. Midpoint: [tex]M = (5,7)[/tex]
4. Slope of perpendicular line: [tex]m_2 = -\frac{3}{4}[/tex]
Step-by-step explanation:
Given
[tex]A = (2,3)[/tex]
[tex]B = (8,11)[/tex]
Solving (1): Slope of AB
Slope (m) is calculated as follows:
[tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]
Where:
[tex]A = (2,3)[/tex] --- [tex](x_1,y_1)[/tex]
[tex]B = (8,11)[/tex] --- [tex](x_2,y_2)[/tex]
So, we have:
[tex]m = \frac{11 - 3}{8 - 2}[/tex]
[tex]m = \frac{8}{6}[/tex]
[tex]m = \frac{4}{3}[/tex]
Solving (2): Length AB
This is solved by calculating the distance of AB using the following formula.
[tex]AB=\sqrt{(x_1-x_2)^2 + (y_1 - y_2)^2}[/tex]
Where:
[tex]A = (2,3)[/tex] --- [tex](x_1,y_1)[/tex]
[tex]B = (8,11)[/tex] --- [tex](x_2,y_2)[/tex]
So:
[tex]AB = \sqrt{(2-8)^2 + (3 - 11)^2}[/tex]
[tex]AB = \sqrt{(-6)^2 + (-8)^2}[/tex]
[tex]AB = \sqrt{36 + 64}[/tex]
[tex]AB = \sqrt{100}[/tex]
[tex]AB = 10[/tex]
Solving (3): Midpoint of AB.
Midpoint, M is calculated as follows:
[tex]M = \frac{1}{2}(x_1+x_2, y_1 + y_2)[/tex]
Where
[tex]A = (2,3)[/tex] --- [tex](x_1,y_1)[/tex]
[tex]B = (8,11)[/tex] --- [tex](x_2,y_2)[/tex]
So:
[tex]M = \frac{1}{2}(2+8,3+11)[/tex]
[tex]M = \frac{1}{2}(10,14)[/tex]
[tex]M = (5,7)[/tex]
Solving (4): Slope of line perpendicular to AB
The relationship between the slopes of two perpendicular lines is:
[tex]m_2 = -\frac{1}{m_1}[/tex]
Where
[tex]m_1[/tex] represents the slope of AB
[tex]m_1 = \frac{4}{3}[/tex]
So:
[tex]m_2 = -1/\frac{4}{3}[/tex]
[tex]m_2 = -1*\frac{3}{4}[/tex]
[tex]m_2 = -\frac{3}{4}[/tex]
