Given the two points (1, 1) and (-1, -1) find the equations of the parallel and perpendicular?

Question

Discuss:

1) What is the nature of the line?

2) What does that mean?

3) What forms are there for lines? And how are these forms described?

4) What is the slope of a horizontal line? A vertical line? How are they described?

5) The y intercept form is the one we use to put our line into the calculator, the standard form we will use soon to solve systems of equations. The point slope form is useful when we know a point and the slope. As in problems like this:

Given the two points (1, 1) and (-1, -1) find the equations of the parallel and perpendicular lines through (-5, 5).

a) What do we do first?

b) Then what?

c) What is the slope of the parallel line?

d) What is the slope of the perpendicular line?

e) Here are some numbers which might be slopes. What are their negative recipricals?

1

1/2

-2

-3/4

Answer 1

Well, you have the points (1,1) and (-1,-1) on the line.  First, I think you need to find the slope of the line given.  

m= (y2 - y1) / (x2 - x1) (forgive my lack of subscript, please)

Inserting what you know, using (1,1) as (x1,y1) and (-1,-1) as (x2,y2)

m= (-1 - 1) / (-1 - 1)

So,

m= (-2) / (-2)

The slope, m, is 1.  Parallel lines have the same slope, so the slope of a line parallel to this one would be m=1.  Perpendicular lines have a slope which is the inverse (proper terminology?) of the original, so the perpendicular line would have a slope of m=-1.  

Next, the equations of the lines.  I think you're looking for y=mx+b.

For the first line, you know m=1.  You also know two points on the line.  So, using one of the given coordinates (I'll use (1,1)), you can find b.

1 = 1 (1) + b  

1 = 1 + b

0 = b

So b, the y-intercept, is 0. The point of interception is (0,0), obviously, but I don't think you need that.

The equation of the line should be y = x.

For the parallel line, you know also that m=1 now.  You have the point (-5, 5) as well.  So, the same way as before:

y = mx + b

5 = 1(-5) + b

5 = -5 + b

10 = b

So, b is 10.  

The equation of the line is y = x + 10.

Finally, the perpendicular line, is m=-1.  You have the point (-5,5).

y = mx + b

5 = -1(-5) + b

5 = 5 + b

0 = b

The y-intercept is 0, and the equation of the line is y = -x.

Finally, I think the only part left is e.  

1 would be -1.  1/2 would be -2.  -2 would be 1/2.  And finally -3/4 is 4/3.  

All you do is take the value, "flip" it, and invert it (positive becomes negative, negative positive).  

Hope that was what you needed.  I know I didn't answer the "Discuss" section; but I assume you can find the answers in your textbook relatively easily, assuming this is for some math related class.