Geometry
Intro to Geometric Constructions
Name___________________________ Per___
For these notes you will need a straightedge (a ruler or protractor will do), a compass, and a
sharpened pencil.
When we draw a shape free-hand, this is called a ________________. If we use
measurement tools, such as a ruler or protractor create a shape, then we say that we
__________________________ the shape.
There is another way to create shapes in geometry. It is a method dating back to the ancient Greeks and other early
geometers. They used only a compass and straightedge to make shapes. These are called _______________________.
The famous Greek mathematician Euclid (325-265 B.C.) established the basic rules for constructions in his most famous
work, Elements.
Technique #1: Duplicating a Segment.
1. Lightly draw a ray from a new point.
2. Use your compass to measure the segment.
3. Without changing your compass, place it on
the endpoint of the ray and swipe a small arc.
4. Mark the new endpoint and connect both
endpoints to make a segment.
Technique #2: Duplicating an Angle.
1. Lightly draw a ray from a new point.
2. Place the compass on the vertex of the angle and swipe
an arc through both rays.
3. Without changing your compass, place it
on the endpoint of the ray and swipe the
same arc.
4. Using your compass, measure the distance between
the intersection points of the arc on the original angle.
5. Duplicate this measurement on the new arc creating
a new point.
6. Draw a ray from the vertex of your new angle through
the new point to complete your duplicated angle.
Application: Duplicate a Shape
Use the techniques you learned above to duplicate the triangle below. Do not erase any construction marks!
Geometry
Geometric Constructions: Part 2
Name_____________________________Per__
Technique #3: Midpoint/Perpendicular Bisector
A perpendicular bisector is a segment or line that passes through the midpoint of another segment and is perpendicular
to that segment.
1. Set your compass to more than half the distance
between the endpoints of the given segment.
2. Place your compass point on one endpoint of the
segment and swing an arc above and below the line.
3. Without changing your compass setting, repeat the
same process from the other endpoint, creating two points
above and below the line segment.
4. Construct a line connecting these two points.
This process constructs a perpendicular bisector and a
midpoint on the original segment.
Application: Median
A median is a segment connecting the vertex of a triangle
to the midpoint of its opposite side. There are three
midpoints and three vertices in every triangle, so every
triangle has three medians.
Application: Midsegment
A midsegment is a segment that connects the midpoints of
two sides of a triangle. Every triangle has three
midsegments.
Geometry
Geometric Constructions: Part 3
Name__________________________ Per ____
Technique #4: Perpendiculars Through Points and Lines
You’ve already learned how to construct a perpendicular bisector or midpoint when given a line segment. But what if you
want to construct a perpendicular line through a given point off the line? Or a given point on the line?
Construct a Perpendicular Line Through A Point Not
On The Line.
1. Set the point of your compass on the given point.
2. Swipe two arcs on the line, creating two points
equidistant from the given point.
3. Use these two points as endpoints of a segment
and now construct a perpendicular bisector. It should
go through the given point not on the line.
Construct A Perpendicular Line Through A Point On
The Line
1. Set the point of your compass on the given point.
2. Swipe two arcs (one on either side of the point) on
the line, creating two points equidistant from the
given point.
3. Use these two points as endpoints of a segment
and now construct a perpendicular bisector. It should
go through the given point on the line.
Application: Altitude (Acute 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.
Application: Altitude (Obtuse Triangle)
Geometry
Geometry Constructions: Part 4
Name_______________________________ Per____
Technique #5: Bisecting an Angle
1. Place the point of your compass on the vertex of the
angle.
2. Swipe an arc that intersects both rays of your angle.
3. Place the point of your compass on one of the new
intersection points and swipe an arc further out between
the two rays. Repeat using the other intersection point,
creating a new point inside your angle.
4. Connect the vertex of the angle to this new point. This
new ray is the angle bisector.
Application: Constructing Specific Angle Measurements/Shapes
1. Use the technique above to construct a 45˚ angle.
2. Construct a regular octagon.
Geometry
Geometry Constructions: Part 5
Technique #5: Constructing Parallel Lines
1. Draw a point that you want your parallel line to
pass through.
2. Draw a line that intersects your point and the
given line. The angle does not matter.
3. Duplicate a(n) corresponding, alternate interior,
or alternate exterior angle from the original line to
the point not on the line.
4. The resulting ray will be your parallel line.
Application: Shapes With Parallel Lines
1. Using the segment below, construct a rhombus.
2. Construct an isosceles trapezoid.
Name______________________________ Per______