3-8: Constructing Parallel and Perpendicular Lines

3-8: Constructing Parallel and Perpendicular Lines

For additional help: Here are some very good animated videos of the constructions:

Parallel Line: http://www.mathopenref.com/constparallel.html

Perpendicular line through a point on the line: http://www.mathopenref.com/constperplinepoint.html

Perpendicular line through a point not on the line: http://www.mathopenref.com/constperpextpoint.html

Parallel Lines:

If you are given a line and a point that is not on the line, you can construct a parallel line to the given line that goes through the given point,

For example, let’s use Line A and point B

1.  Create a point anywhere line A and label it C.

Then create line BC.

2.  Put the point of your compass on C and draw an arc.

3.  Don’t change the compass setting! Keep it the same as what you used in step 2.

Put the compass point on point B. Draw an ark.

4.  Put your compass point on this intersection:

And stretch out your compass to this intersection.

5.  Don’t change the compass setting you got from step 4. Put the compass point on this intersection:

And draw an ark.

Let’s label the resulting intersection D.

6.  Create line BD.

Line BD is parallel to line A because of the Corresponding Angles Postulate. You have created two equal angles along a transversal.

Perpendicular Lines (There are two different kinds of constructions relating to these)

Perpendicular at a Point on a Line

If you are given a line and a point on the line, you can construct a perpendicular line to the given line that goes through the given point.

For Example, let’s use line A and point B.

1.  Put the compass point on point B and draw two arks on line A using the same compass settings. Let’s label these intersections X and Y.

2.  Stretch out the compass so that it is wider than the distance between point X and point B.

Put the compass on point X and draw an ark above line A.

3.  Don’t change the compass setting from step 2. Place the compass point on point Y and draw an ark above line A.

It intersects the ark from step 2. Let’s label this intersection Z.

4.  Draw line BZ. It is perpendicular to line A.

Perpendicular from a Point to a Line

If you are given a line and a point that isn’t on the line, you can construct a perpendicular line to the given line that goes through the given point.

For example, let’s use line X and point Y.

1.  Stretch out your compass so that it is wider than the distance between point Y and line X. Put your compass point on Y and draw two arks on line X without changing the compass settings. Let’s label these intersections A and B.

2.  Put your compass point on A and draw an ark under line X.

3.  Don’t change the compass setting from step 2. Put your compass point on B and draw an ark under line X. It intersects the ark made in step 2. Let’s label this intersection Z.

4.  Draw line YZ. It is perpendicular to line X.

Line YZ shows the shortest distance possible between line X and point Y.

Review Questions

1.  Given: Line GH. Point J is not on the line.

Draw a line that is parallel to line GH and goes through J.

2.  Given: Line OP. Point W is on the line.

Draw a line that is perpendicular to line OP and goes through W.

3.  Given: Line SR. Point E is not on the line.

Draw a line that is perpendicular to line SR and goes though E.

4.  What is the shortest distance between line SR and point E from problem 3?

5.  Create quadrilateral ABCD with one pair of parallel sides:

Create line AB and point D.

Then draw a line that is parallel to line AB and goes though point D. Label that parallel line CD. Then connect the dots to form a quadrilateral.