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### Find Slope, Equation, Midpoint

```Date: 06/04/2003 at 23:50:08
From: Ashley
Subject: Slopes

A = (-4,6) and B = (-2,1)

1) find the slope;

2) find the equation of the line containing A and B;

3) find the coordinates for the midpoint AB;

4) find the distance between A and B.
```

```
Date: 06/05/2003 at 09:57:01
From: Doctor Dotty
Subject: Re: Slopes

Hi Ashley,

Thanks for the question.

Any (non-vertical) line can be represented by the equation:

y - y1 = m(x - x1)

Where m is the gradient (slope) and two points it passes through are
(x, y) and (x1, y1).

Let's think about a line passing through (2, 4) and (-3, 6) as an
example.

We have two points, so x = 2, y = 4; x1 = -3, y1 = 6.

This gives:

4 - 6 = m(2 - -3)

-2 = 5m

m = -2/5

So the gradient (slope) is -2/5.

Next we need an equation of the line. We can find this using one point

y - y1 = m(x - x1)

y - 6 = (-2/5)(x - -3)

y - 6 = (-2/5)(x + 3)

Multiply by 5:

5y - 30 = -2(x + 3)

5y - 30 = -2x - 6

5y + 2x - 24 = 0

Which is the equation of the line.

Now, the midpoint.

That is the point that lies horizontally halfway between x and x1, and
vertically halfway between y and y1.

To remind us: x = 2, y = 4; x1 = -3, y1 = 6.

Halfway between 2 and -3 is -0.5.

Halfway between 4 and 6 is 5.

Therefore the coordinates of the midpoint are (-0.5, 5).

For easier use in the future, you can write this method as:

x + x1     y + y1
Midpoint is at   ------  ,  ------
2          2

Now, the distance from A to B.

.               |
.           |
A      6 +
|    .   |
|        |
|        |  .
|_ _ _ 4_+_ _ _.B
|        .
|           .
|              .
|                 .
-----------+--------+------+---------------
-3        |      2               .
|
|

Let's look at that triangle:

A
|    .   d
2|
|_          .
|_|_ _ _ _ _ _ .B
5

Using Pythagoras' theorem,

2^2 + 5^2 = d^2             (where ^2 means squared)

4 + 25 = d^2

sqrt(29) = d

Which is the distance between the two points.

For easier use in the future, you can write this method as an
equation:

d = sqrt[ (x - x1)^2  +  (y - y1)^2 ]

Can you apply this to your question now?

Write back if I can be of any more help - on this or anything else.

- Doctor Dotty, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Linear Equations

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