Question

A line segment AB has the coordinates A (2,3) AND B ( 8,11) answer the following questions (1) What is the slope of AB? (2) What is the length of AB? (3) What are the coordinates of the mid point of AB?(4) What is the slope of a line perpendicular to AB ?

Answer 1

1. Slope:
`m = (4)/(3)`

2. Distance:
`AB = 10`

3. Midpoint:
`M = (5,7)`

4. Slope of perpendicular line:
`m_2 = -(3)/(4)`

Explanation:

Given

`A = (2,3)`

`B = (8,11)`

Solving (1): Slope of AB

Slope (m) is calculated as follows:

`m = (y_2 - y_1)/(x_2 - x_1)`

Where:

`A = (2,3)` ---
`(x_1,y_1)`

`B = (8,11)` ---
`(x_2,y_2)`

So, we have:

`m = (11 - 3)/(8 - 2)`

`m = (8)/(6)`

`m = (4)/(3)`

Solving (2): Length AB

This is solved by calculating the distance of AB using the following formula.

`AB=√((x_1-x_2)^2 + (y_1 - y_2)^2)`

Where:

`A = (2,3)` ---
`(x_1,y_1)`

`B = (8,11)` ---
`(x_2,y_2)`

So:

`AB = √((2-8)^2 + (3 - 11)^2)`

`AB = √((-6)^2 + (-8)^2)`

`AB = √(36 + 64)`

`AB = √(100)`

`AB = 10`

Solving (3): Midpoint of AB.

Midpoint, M is calculated as follows:

`M = (1)/(2)(x_1+x_2, y_1 + y_2)`

Where

`A = (2,3)` ---
`(x_1,y_1)`

`B = (8,11)` ---
`(x_2,y_2)`

So:

`M = (1)/(2)(2+8,3+11)`

`M = (1)/(2)(10,14)`

`M = (5,7)`

Solving (4): Slope of line perpendicular to AB

The relationship between the slopes of two perpendicular lines is:

`m_2 = -(1)/(m_1)`

Where

`m_1` represents the slope of AB

`m_1 = (4)/(3)`

So:

`m_2 = -1/(4)/(3)`

`m_2 = -1*(3)/(4)`

`m_2 = -(3)/(4)`