Video: Finding the Equation of a Perpendicular Line

Given 𝐴 (4, 4) and 𝐡 (2, βˆ’4), find the equation of the perpendicular to 𝐴𝐡 that passes through the midpoint of this line segment. Give your answer in the form 𝑦 = π‘šπ‘₯ + 𝑐.

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Video Transcript

Given 𝐴 four, four and 𝐡 two, negative four, find the equation of the perpendicular to 𝐴𝐡 that passes through the midpoint of this line segment. Give your answer in the form 𝑦 equals π‘šπ‘₯ plus 𝑐.

So we’ve been asked to find the equation of this line in the form 𝑦 equals π‘šπ‘₯ plus 𝑐, which is the slope-intercept form of the equation of a straight line. In order to find the equation of this line, we need to know two things: firstly, the slope of the line and secondly, the coordinates of any point that lies on the line. So let’s think about how to calculate each of these things, beginning with the slope.

We’re told first of all that this line is perpendicular to the line segment 𝐴𝐡. Now, this is important information because a key fact about perpendicular lines is that their slopes multiply to negative one. In other words, they are the negative reciprocals of each other.

If we think of the two slopes of perpendicular lines as π‘š one and π‘š two, then their product is equal to negative one. But also, π‘š one is equal to negative one over π‘š two and π‘š two is equal to negative one over π‘š one. So if we know the slope of one of the lines, we can calculate the slope of its perpendicular line by dividing negative one by the value we know.

As we’ve been given the coordinates of both 𝐴 and 𝐡, we can calculate the slope of the line segment 𝐴𝐡 first of all. The formula that we can use is 𝑦 two minus 𝑦 one over π‘₯ two minus π‘₯ one, where π‘₯ one, 𝑦 one and π‘₯ two, 𝑦 two are the coordinates of two points on the line.

So if we think of 𝐴 as the point π‘₯ one, 𝑦 one and 𝐡 as the point π‘₯ two, 𝑦 two, then this gives negative four minus four over two minus four. If you think of 𝐴 and 𝐡 the other way round, so 𝐴 as π‘₯ two, 𝑦 two and 𝐡 as π‘₯ one, 𝑦 one, it doesn’t matter; you’re still get the same value for the slope.

Now, the numerator is negative four minus four which is negative eight and the denominator is two minus four which is negative two. A negative divided by a negative gives a positive answer and eight divided by two is four. So the slope of the line segment 𝐴𝐡 is four.

Now, to find the slope of the perpendicular line, remember that we need to find the negative reciprocal of this value. So we need to divide negative one by four and it gives negative a quarter. So we found the slope of our line. And now, we need to think about finding the coordinates of one point that lies on the line.

We’re told in the question that our line passes through the midpoint of this line segment β€” so that’s the line segment 𝐴𝐡. We also have a formula for calculating the midpoint of a line segment. If the two points are again thought of as π‘₯ one, 𝑦 one and π‘₯ two, 𝑦 two, then the π‘₯-coordinates to the midpoint is π‘₯ one plus π‘₯ two over two and the 𝑦-coordinate to the midpoint is 𝑦 one plus 𝑦 two over two.

This just means that the π‘₯-coordinate to the midpoint is the average or mean of the π‘₯-coordinates and the 𝑦-coordinate to the midpoint is the average of the 𝑦-coordinates.

Substituting the coordinates of 𝐴 and 𝐡 gives four plus two over two for the π‘₯-coordinate and four plus negative four over two for the 𝑦-coordinate. This gives six over two which is three and zero over two which is zero. So the midpoint of the line segment 𝐴𝐡 is three, zero.

So now that we know both the slope and the coordinates of one point on our line, we can find its equation. Remember we’re asked for the equation in the form 𝑦 equals π‘šπ‘₯ plus 𝑐, where π‘š represents the slope of the line. We can, therefore, substitute the value that we’ve calculated for the slope of the line straightaway. And it gives 𝑦 is equal to negative one-quarter π‘₯ plus 𝑐.

In order to find the value of 𝑐, we need to use the fact that the point with coordinates three, zero lies on the line, which means that when π‘₯ is equal to three, 𝑦 is equal to zero. And this pair of values satisfy the equation of the line. So by substituting π‘₯ equals three and 𝑦 equals zero into the equation of our line, we now have the equation zero is equal to negative a quarter multiplied by three plus 𝑐. And this is an equation that we can solve to find the value of 𝑐.

Negative a quarter multiplied by three is negative three-quarters. So we have zero is equal to negative three quarters plus 𝑐. To solve for 𝑐, we need to add three quarters to each side of the equation. And it gives three-quarters is equal to 𝑐.

Finally, then, we just need to substitute this value 𝑐 into the equation of our line. The equation of the line perpendicular to 𝐴𝐡 and passing through its midpoint is 𝑦 equals negative a quarter π‘₯ plus three-quarters.

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