Warmup Questions II

These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.

Warmup Question 3

Find the equation of the line passing through the points $(2,5)$ and $(8,-3)$.

Show Answer 3

Suppose $(x_1,y_1)=(2,5)$ and $(x_2,y_2)=(8,-3)$. The slope of the line is

$$m = \dfrac{y_2-y_1}{x_2-x_1}=\dfrac{-3-5}{8-2} = \dfrac{-8}{6} = -\dfrac{4}{3}.$$

Using the point-slope equation, we obtain

$$y-y_1=m(x-x_1)$$

$$y-5=-\dfrac{4}{3}(x-2)$$

$$y = -\dfrac{4}{3}x-\dfrac{4}{3}\cdot 2 +5$$

$$y = -\dfrac{4}{3}x+\dfrac{11}{3}$$

Warmup Question 4

What is the slope of the perpendicular line to a line whose slope is $-3$?

Show Answer 4

$\dfrac{1}{3}$

Review II

If you are not comfortable with the Warmup Questions, don’t give up! Click on the indicated lesson for a quick catchup. A brief review will help you boost your confidence to start the new lesson, and that’s perfectly fine.

Need a review? Check Lesson ??

Quick Intro II

This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.

A Quick Intro to Perpendicular Bisector

Key Words. Perpendicular bisector, line segment, slope, slope of the perpendicular line, midpoint, equation of the line.

The perpendicular bisector of a line segment connecting $P_1$ and $P_2$ is the line perpendicular to this line segment that passes through the midpoint of $P_1$ and $P_2$.

On the graph below, $P_1$ and $P_2$ are the points in red, the line segment is in purple, the midpoint is in black, and the perpendicular bisector is in green.

To find the equation of the perpendicular bisector of the line segment connecting $P_1$ and $P_2$:

  • Find the midpoint of $P_1$ and $P_2$. Call it $M = (x_1,y_1)$.
  • Find the slope of the line passing through $P_1$ and $P_2$. Call it $s$.
  • The slope of the perpendicular line to the line from the previous step is $m = -\dfrac{1}{s}$
  • The perpendicular bisector is the line of slope $\bar m$ that passes through $M=(x_1,y_1)$. The equation is $$y-y_1 = m(x-x_1).$$

Video Lesson II

Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!

Video Lesson 2

A video lesson on Perpendicular Bisector [2:50]

A description of the video

In this video you will see how to find the perpendicular bisector passing through $(1,2)$ and $(5,4)$.

Try Questions II

Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.

Try Question 4

Find an equation for the perpendicular bisector of the line segment joining $(1,4)$ and $(-5,2)$.

Show Answer 4

The midpoint is $$M=\left(\dfrac{1+(-5)}{2},\dfrac{4+2}{2}\right) = (-2,3).$$ 

The slope of the line passing through $(1,4)$ and $(-5,2)$ is $m =\dfrac{2-4}{-5-1} = \dfrac{-2}{-6} = \dfrac{1}{3}$.  The slope of the perpendicular line is $-3$. 

The equation of the perpendicular bisector passing through $(-2,3)$ is $$ y -3 = -3(x-(-2)) \quad\Longrightarrow\quad y – 3 = -3x-6  \quad\Longrightarrow\quad y = -3x-3.$$

The blue line is the perpendicular bisector on the graph.