Perpendicular Line Constructions

Lindsay ChenowethJonathan Bojarski

Introduction What is a construction?

A “drawing” using only a compass and straightedge that renders “exact” results.

Construction is not simply drawing, it is like building a drawing.

No instrument or surface can actually create exact results.

Why do we need constructions?

Perpendicular Lines Perpendicular lines: two lines that intersect,

and at their intersection form four angles that equal 90 degrees each.

Perpendicular Lines 3 Constructions of Perpendicular Lines:

Given two points A and B on a line, construct any line perpendicular to the given line.

Given a line and a point A on that line, construct a perpendicular line through point A.

Given a line and a point A off of that line, construct a line from point A perpendicular to the given line.

Construction 1 Given two points on a line, construct any

line perpendicular to the given line.

A

B

Construction 1 Step 1: Draw a circle centered at point A.

Construction 1 Step 2: Draw a circle with the same radius

centered at point B.

Construction 1 Step 3: Draw points at the places where the

two circles intersect.

Construction 1 Step 4: Draw a line that connects the two

points. This line should be perpendicular to line AB.

Construction 2 Given a line and a point A on that line,

construct a line perpendicular to that line through A.

Construction 2 Step 1: Draw a circle of any radius centered

at point A.

Construction 2 Step 2: Draw points at both places where

the circle intersects the given line. Name these points B and C.

Construction 2 Step 3: Draw a circle centered at point B

with a radius larger than segment BA. Draw a circle centered at point C with the same radius as circle B.

Construction 2 Step 4: Draw points at both places where

circles B and C intersect. Connect these points with a line. This line should be perpendicular to line BAC through point A.

Construction 3 Given a line and a point A off of that line,

construct a line perpendicular to the given line that runs through point A.

Construction 3 Step 1: Draw a circle centered at point A

that intersects the given line in two places. Draw points where the circle intersects the line. Name these points B and C.

Construction 3 Step 2: Draw a circle centered at point B

with a radius greater than or equal to BA. Draw a circle centered at point C with a radius of CA.

Construction 3 Step 3: Draw points where these circles

intersect. Draw a line that connects this point with point A. This line should be perpendicular to line BC.

http://www.mathsnet.net/campus/construction/i_index.html

Constructions Online

A3: construction 1

B4: construction 3B2: construction 2

Question Using what you know about the different

ways to construct perpendicular lines, can you come up with a way to construct parallel lines?