Write an equation of the line perpendicular to the given line that contains P. p(4,4);y=-2x-8

Answer 1

Hi there Michael,

So here you are given a slope in standard form:

y = -2x - 8

The -2 means that for every 1 change in x, y will change by -2

The -8 means that 8 is subtracted from whatever value y would have had before

This also means that when x = 0, -2x = 0, leaving just -8 which gives us the y-intercept at (0,-8)

To find a line that is perpendicular to this line you have to think about the direction is traveling in on the coordinate plane. A slope of -2 means that as x increases y goes down by 2, meaning it has a negative slope that travels down into the 4th quadrant as x gets bigger. Well, if you have a negative slope, to form a 90-degree angle, a line would have to have a positive slope, or up into the 1st quadrant as x gets bigger.

This tells us the first important part: The perpendicular slope must have the opposite sign of the 1st slope

Now, you may think that the perpendicular slope would be 2, but that is not quite right. It may not be so easy to imagine in this example, but imagine if the slope was 10 and we tried to use a slope of -10 to form a perpendicular line. The line with a positive 10 slope would be travel steeply upwards while the line with the negative 10 slope would travel sharply downwards. You can imagine that these lines would not meet to form a 90-degree angle at their intersection.

As it turns out, the slope of a line perpendicular to another is the negative reciprocal.

In this case, the negative reciprocal of -2 is 1/2

Now that we have the slope we can write the equation

y = x/2 + b

substituting the x and y values (4,4) we have

4 = 4/2 + b

solve for b

b = 2

This gives us the final equation of

y = x/2 + 2

Answer 2

Write an equation of the line perpendicular to the given line that contains P. p(4,4);y=-2x-8

The required line must satisfy the perpendicular condition m₁m₂=-1 and must also pass through the given point (4,4).

We start by finding the gradient (m₁) of the line whose equation is given:

Since the equation is already in slope-intercept form, we get m₁ = -2 (by observation).

Next we find the equation of the perpendicular line using the expression m₁m₂=-1. So that m2 = -1/-2 = 1/2

Now, to find the value of intercept (b) for the line whose gradient is 1/2 and which passes through the point (4,4) we have

4 = 1/2 x 4 + b or b = 2

Finally we have the equation of the required line as y = 1/2x + 2