Chapter 3 Parallel and Perpendicular Lines
#1) Construct a Perpendicular Line Through a Point
Goal: Given a line and a point on the line, construct a perpendicular line through the given point.
Step 1: Find two points on the line that are equidistant from the given point.
- Select "Circle with Center through Point" in the 4th dropdown window.
- Click on P to center the circle at P, then click anywhere on the line to the left of P. You now have a circle centered at P, through point A.
- Select the "Point" tool (first option in the the 2nd dropdown window). Click on where the circle intersects the line, on the right side of P.
**Whenever you place a point at an intersection, double check it that it shows up as black.
If the point shows up as blue, then it is not on the intersection.**
You should now have points A and B as the two intersection points.
These two points on the line are equidistant from P, because they are on the same circle centered at P.
Step 2: Create two circles with the same radius, one centered at A and one centered at B.
- Use the "Circle with Center through Point" tool to make a circle, centered at A, that is a little bit larger than the circle centered at P. You should notice a point on A labelled "C."
- Select the "Compass" tool in the 4th dropdown window. Click on A and then C - this "fixes" the radius of your new circle so that it equals AC.
- Now click on point B.
You should now have two intersecting circles, one centered at B, and one centered at A, that have the same radius.
Step 3: Locate the intersection of the two circles.
- Use the "POINT" tool to label the point above P where the two large circles intersect. It should show up as point D.
Step 4: Construct a line through points D and P.
- Use the "Line" tool (first tool in the 3rd dropdown window), then click on points D and P.
Step 5: "lighten" the arc marks.
- Click “Move” (the 1st window) to allow you to click and drag objects.
- Click on one of the circles.
- Notice a menu just below the upper right corner. It has a circle and a triangle over three gray lines. Click on this menu to expand the color/outline options.
- Click on the "solid line segment" box. A dropdown menu of dotted lines will show up. Select the fourth "dotted line" option to make the circle have a thin, dotted outline.
- Repeat this for all of your circles.
Now, the two lines should stand out visually, but the circles are still visible as a way to show your work.
Perpendicular Line through a Point
Move points P, A, and C around. As you move around the points, the circles change size and the lines rotate. Still, what special relationship between the two lines appears to always be true?
#2) Construct a Parallel Line through a Point
Given a line and a point NOT on the line, we can construct a parallel line through this point using the theorem that two lines perpendicular to the same line are parallel.
Step 1: Construct a perpendicular line through the given point.
a) Find two points on the line that are equidistant from point P.
- Use the "Construct Circle with Center through Point" tool --> Click on P --> Click on the line so that you have a circle that intersects the line at point C AND another point.
- Use the "Point" tool to label the other intersection point (it should show up as point D).
b) Find a new point below the line that is equidistant from the two points you just found.
- Use the "Construct Circle with Center through Point" tool to create a circle centered at D. A new point E should show up on the circle.
- Select the "Compass" tool. Click on point D & E to fix the radius, then center the circle at C.
- Now, you should have two circles with the same radius, one centered at D and one centered at C.
- Use the "Point" tool to label the intersection of these two circles. It should show up as point G. If they don't intersect, use the "Move" tool to click and drag point E until the circles are big enough to intersect.
c) Construct a line through P and F.
- Use the "Line" tool, then click both points.
** Pause: Color-code before the next part **
- Make the color of the perpendicular line you just constructed red.
- Make the circles have dotted outlines so that the lines you constructed show up more clearly.
- Pro-tip: you can hold the "CTRL" button while you click in order to select multiple objects at once.
Confirm your work is correct so far: If you click and drag either of the blue endpoints of the original segment, the red line should remain perpendicular to it.
Step 2: Construct a new line that is perpendicular to the red line, through point P.
We'll use the same process as in problem #1:
- Construct a circle centered at P to find and label two points on the red line that are equidistant from P.
- Construct a circle centered at one of these new points.
- Use the "Compass" tool to copy a circle with the same-size radius, centered at the other new point.
- Place one last point on the intersection of these congruent circles, then construct a line through this point and point P.
The original line and the new line will be parallel, because they are both perpendicular to the same line!
You can confirm this is correct by clicking and dragging any blue points on the screen.
Parallel Line through a Given Point
#3) Construct a Rectangle, When Given Two Side Lengths (Strategy #1)
A rectangle is a quadrilateral with four sets of perpendicular sides.
You can construct a rectangle by constructing perpendiculars, as you did in the previous two tasks.
In this task, you will build a rectangle with width DC and height XY.
Step 1: Construct a perpendicular line through D, using the process shown in task #1.
Make this line red.
Make your circles have a thin dotted outline, and drag the points to make the circles smaller, before you move on (just to clear up the screen).
Step 2: Construct a perpendicular line through C, using the process shown in task #1.
Make this line orange.
Make your circles have a thin dotted outline, and drag the points to make the circles smaller, before you move on (just to clear up the screen).
Step 3: Locate and label point A by using the "Compass" tool to copy the correct length (make a circle with radius XY, centered at D).
To rename a point: right click the point --> click on "rename."
Step 4: Construct a perpendicular line through A, using the process shown in task #1.
Make this line blue.
Step 5: Locate and label point B as the intersection of the sides.
We know that ABCD is a rectangle, because all of its sides are perpendicular by construction.
Clean up your work by making your circles dotted + dragging the points so that the circles are not too big.
Check your work:
- Click and move around points D and C to check that the sides remain perpendicular.
- Click and move around X and Y to check that the height of the rectangle stays equal to XY.
Construct a Rectangle, Given Two Side Lengths (#1)
#4) Construct a Rectangle, When Given Two Side Lengths (Strategy #2)
In this task, you will build the same rectangle from the same given information. However, this will show you with an easier way to do it.
You can use the fact that opposite sides of a rectangle are equal in length to construct the rectangle, rather than constructing multiple different perpendiculars.
Step 1: Construct a perpendicular line through D, using the process shown in task #1.
Make this line red.
Make your circles have a thin dotted outline, and drag the points to make the circles smaller, before you move on (just to clear up the screen).
Step 2: Locate and label (rename) point A by using the "Compass" tool to copy the correct length (make a circle with radius XY, centered at D).
Step 3: Use the "Compass" tool to create a circle with radius CD, centered at A. Every point on this circle represents a point that is "CD" units from point A.
Step 4: Use the "Compass tool to create a circle with radius XY, centered at C. Every point on this circle represents a point that is "XY" units from point C.
Step 5: Locate and label (rename) point B as the intersection of the circles. This is the only point that is exactly "XY units from point C AND "DC" units from point A.
This time, we know that ABCD is a rectangle, because its opposite sides have the same length, and it contains a perpendicular set of sides.
Clean up your work by making your circles dotted + dragging the points so that the circles are not too big.
Check your work:
- Click and move around points D and C to check that the sides remain perpendicular.
- Click and move around X and Y to check that the height of the rectangle stays equal to XY.
Construct a Rectangle, Given Two Side Lengths (#2)
In what two ways are opposite sides of a rectangle related to each other?
#5) Diagonals of a Rectangle
a) Use the "Segment" tool to add the diagonals in rectangle ABCD below.
b) Place a point at the intersection of these diagonals.
Move around points A, C, and D so the rectangle and its diagonals change. As you do so, investigate: What relationship(s) between the diagonals never seem to change, no matter the dimensions of the rectangle?
Diagonals of a Rectangle
In what two ways are the diagonals of a rectangle related to each other?
#6) Construct a Square
A square is a quadrilateral in which all four sides are congruent, and adjacent sides are perpendicular.
In other words, a square is an equilateral rectangle.
Using the skills from this assignment, construct a square from the given side length, shown below.