1. Constructions Using Tools in Geometry
Lesson 3.1 – 3.5
Constructions Using Tools in Geometry
The compass and the straightedge have been useful tools in geometry for thousands of years.
By tradition, neither a ruler nor a protractor is ever used to perform geometric
constructions, because no matter how precise we try to be, measurement always
involves some amount of inaccuracy.
Rulers and protractors are measuring tools, not construction tools. You may use a ruler as a straightedge in constructions, provided you do not use its marks for measuring.
We will learn how to construct the following using these two tools:
Duplicating Line Segments Copy a line segment
Duplicating Angles Copy an angle
Constructing Equilateral Triangles Equilateral Triangle
Constructing Perpendicular Bisector Perpendicular bisector of a line segment
Perpendicular to a line Perpendicular to a line from an external point
Angle Bisectors Bisect an angle
Parallel Lines A parallel to a line through a point (angle copy method)
JRLeon Geometry Chapter 3.1 – HGSH

2. Constructions Using Tools in Geometry
Lesson 3.1 – 3.5
Constructions Using Tools in Geometry
Perpendicular Bisector Perpendicular bisector of a line segment
Constructing the perpendicular bisector also locates the midpoint of a segment.
Now that you know how to construct the perpendicular bisector and the midpoint, you can construct rectangles, squares, and right triangles.
You can also construct two special segments in any triangle: medians and midsegments.
Median :
is the segment connecting the vertex of a triangle to the midpoint of its opposite side.
There are three midpoints and three vertices in every triangle.
Every triangle has three medians.
Midsegment:
is the segment that connects the midpoints of two sides of a triangle.
A triangle has three sides, each with its own midpoint,
Every triangle has three midsegments.
JRLeon Geometry Chapter 3.1 – HGSH

3. Constructions Using Tools in Geometry
Lesson 3.1 – 3.5
Constructions Using Tools in Geometry
Perpendicular from a point to a line Perpendicular to a line from an external point
The construction of a perpendicular from a point to a line lets you find the shortest distance from a point
to a line.
The geometry definition of distance from a point to a line is based on this construction, and it
reads,
“The distance from a point to a line is the length of the perpendicular segment from the point to the line.”
You can also use this construction to find an altitude of a triangle.
An altitude of a triangle is a perpendicular segment from a vertex to the opposite side or to a line containing the opposite side.
If you are in a room, look over at one of the walls.What is the distance from where you are to that wall? How would you measure that distance?
There are a lot of distances from where you are to the wall, but in geometry when we speak of a distance from a point to a line we mean the perpendicular distance.
JRLeon Geometry Chapter 3.1 – HGSH

4. Constructions Using Tools in Geometry
Lesson 3.1 – 3.5
Constructions Using Tools in Geometry
Constructing Angle Bisectors
On a softball field, the pitcher’s mound is the same distance from each foul line.
So it lies on the angle bisector of the angle formed by the foul lines.
As with a perpendicular bisector of a segment, an angle bisector forms a line.
Angle Bisector Construction Bisect an angle
On a softball field, the pitcher’s mound is the same distance from each foul line, so it lies on the angle bisector of the angle formed by the foul lines. As with a perpendicular bisector of a segment, an angle bisector forms a line
of symmetry.
JRLeon Geometry Chapter 3.1 – HGSH

5. Constructions Using Tools in Geometry
Lesson 3.1 – 3.5
Constructions Using Tools in Geometry
Constructing Parallel Lines
Parallel lines are lines that lie in the same plane and do not intersect.
Parallel Lines Construction A parallel to a line through a point (angle copy method)
JRLeon Geometry Chapter 3.1 – HGSH

6. Constructions Using Tools in Geometry
Lesson 3.1 – 3.5
Constructions Using Tools in Geometry
Slopes of Parallel and Perpendicular Lines
If two lines are parallel, how do their slopes compare?
If two lines are perpendicular, how do their slopes compare?
If the slopes of two or more distinct lines are equal, are the lines parallel?
JRLeon Geometry Chapter 3.1 – HGSH

7. Constructions Using Tools in Geometry
Lesson 3.1 – 3.5
Constructions Using Tools in Geometry
Slopes of Parallel and Perpendicular Lines
If two lines are parallel, how do their slopes compare?
If two lines are perpendicular, how do their slopes compare?
If two lines are perpendicular, their
slope triangles have a different
relationship. Study the slopes of the
two perpendicular lines at right.
JRLeon Geometry Chapter 3.1 – HGSH