Transcript Constructions Day 1 Now for some vocabulary                    Angle: the union of two distinct rays that have a common end point Vertex: the common end.

Constructions
Day 1
Now for some vocabulary
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Angle: the union of two distinct rays that have a common end point
Vertex: the common end point of an angle
Line segment: the part of a line between two points on the line
Ray: a ray begins at a point and goes on forever in one direction
Intersecting lines: two lines meet at a point. That point is the only point that belongs to both
lines
Intersection: all the points 2 figures have in common
Parallel lines: two lines in a plane that do not intersect
End points: the points at the end of a line segment
Compass: instrument used to draw circles. It has two tips – center and pencil.
Bisector: divides a segment or angle into two equal parts
Perpendicular lines: two lines that intersect to form four right angles
Circle: all points in a plane at a constant distance, called the radius, from a fixed point, called
the center
Radius: the length of a line segment between the center and circumference of a circle
Equidistant: equally distant from any two or more points
Arc: a continuous portion of a circle
Congruent: having the same size, shape and measure 
Point: One of the basic undefined terms of geometry. No length, width, or thickness. Dot
represents it.
Line: One of the basic undefined terms of geometry. A set of points whose length goes on
forever in two opposite directions.
Plane: One of the basic undefined terms of geometry. Goes on forever in all directions; twodimensional and flat.
Are you ready to move?
Stand up, push your chair under and
face me.
We are going to use our arms to
show our vocabulary terms we just
learned.
Explore
► Please
take some white paper, a straight
edge and a compass.
► Practice making lines, circles, arcs, and
shapes if you can.
► Make sure you get used to the compass.
What are Constructions?
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The study of Geometry was born in Ancient Greece, where mathematics was thought
to be embedded in everything from music to art to the governing of the universe.
Plato, an ancient philosopher and teacher, had the statement, “Let no man ignorant
of geometry enter here,” placed at the entrance of his school. This illustrates the
importance of the study of shapes and logic during that era. Everyone who learned
geometry was challenged to construct geometric objects using two simple tools,
known as Euclidean tools:
 A straight edge without any markings
 A compass
 The straight edge could be used to construct lines; the compass to construct circles.
As geometry grew in popularity, math students and mathematicians would challenge
each other to create constructions using only these two tools. Some constructions
were fairly easy (Can you construct a square?), some more challenging, (Can you
construct a regular pentagon?), and some impossible even for the greatest
geometers (Can you trisect an angle? In other words, can you divide an angle into
three equal angles?). Archimedes (287-212 B.C.E.) came close to solving the
trisection problem, but his solution used a marked straight edge.
Challenge
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Your First Challenge: Can you construct a Target?
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A very simple target consists of three circles.
The largest circle would have a radius that is three
times the length of the radius of the smallest circle
The middle circle would have a radius two times the
length of the radius of the smallest circle.
On a separate sheet of paper, construct a target with a
straight edge and compass.
Then write a set of instructions that another student
could use to create your target.
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Constructions
Day 2
Copying a Line Segment
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Begin with a line segment AB.
Draw a line with a straightedge
longer than segment AB below.
Label the left endpoint C .
Place the compass point on point
A.
Stretch the compass so that the
pencil is exactly on point B.
Without adjusting the compass
span, place the compass point on
point C and swing the pencil so
that it intersects the line.
Label the intersection point D.
A
B
C
C
D
Copying an Angle
Begin with CAB.
Draw another line and label the left endpoint A’.
With the compass on point A, stretch its width to point B.
Without adjusting the compass, place the compass point on A’ and
draw a wide arc across the line. This establishes a new point B’.
Place the compass point on B and stretch its width to point C.
Without changing the compass span, place the compass point on
point B’ and draw an arc across the previously drawn arc.
The intersection of these two arcs becomes point C’.
Using a straightedge, draw a line from point A’ to C’ where the arcs
intersect.
 C’A’B’ is  (equal in measure) to  CAB.
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C`
C
A
A`
B
B`
Constructions
Day 3
Bisecting a Segment
• Begin with a line segment AB.
• Place the compass point on point A.
• Stretch the compass along segment AB to a length greater
than half the segment length.
• Construct a circle (or wide arc) with center at point A.
• Without adjusting the compass span, place the compass
point on point B.
• Construct a circle (or wide arc) with center at point B.
• Mark and label the intersection points of the two circles as
points C and D.
• Using a straightedge, draw a line through points C and D.
C
• Line CD intersects
line AB at the midpoint, M.
M
A
B
D
Perpendicular Bisector Student
Investigation
• Draw a point on the perpendicular bisector.
• Measure the distance between the point on the
perpendicular bisector and one endpoint of the
segment.
• Compare this distance to the distance between
the point on the perpendicular bisector and the
other segment endpoint.
Question for Students:
• What is true about the distances from a point on
the perpendicular bisector to each endpoint of a
segment?
Angle Bisector
• Begin with an angle.
• Draw a circle (or wide arc) at point O with an arbitrary
radius, making certain the circle intersects both angle
sides.
• Label the points on the angle sides as A and B.
• Draw a circle (or wide arc) at point A such that its radius is
more than half the distance between A and B.
• Without adjusting the compass span, place the compass
point on B and draw a circle (or wide arc).
• Mark and label the intersection point of the two arcs as
point C.
• Using a straightedge, draw a line through points O and C.
A
O
B
C
Angle Bisector Student
Investigation
Question for Students:
• What is true about the distances from a
point on an angle bisector to each side of
the angle?
Constructions
Day 4
Perpendicular to a Line Through a Point on
a Line
• Begin by drawing line segment.
• Draw and label a point C anywhere on
the line but not too close to either end.
• Construct a circle (wide arc) with center
at point C that intersects the line.
• Where the circle intersects the line,
label the intersection points as A and B.
• Place the compass point on point A and
stretch the compass to a distance
greater than AC.
A
• Draw an arc above the line.
• Without adjusting the compass span,
place the compass point on B and draw
an arc above the line, generating
intersecting arcs.
• Label the point D.
• Construct the perpendicular bisector of
segment AB.
D
C
B
Perpendicular to a Line Through an
External Point
• Begin with a line and a point C not
on the line.
• Construct a circle (or wide arc) with
center at point C and radius greater
than the distance from C to the line.
• Where the circle intersects the line,
label the intersection points as A
and B.
• Place the compass point on A and
draw an arc below the line.
• Without adjusting the compass
span, place the compass point on
B and draw an arc, generating
intersecting arcs.
• Label the intersecting point D.
• Construct a perpendicular bisector
of segment AB.
C
A
B
D
Constructions
Day 5
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Parallel
Line
through
a
Given
Point
Begin with a line segment and a point A not on
the line.
Draw a line from A through the original line
label the intersection X.
Put the point of your compass on X and draw
and arc that intersects both lines, label the
intersection points B and C.
Keep the compass width the same and make
the same arc with the compass point at A label
the intersection point D.
With the compass point on B measure from B
to C.
Keep the same width, place the point of the
compass on D and make an arc that intersects
the present arc label the intersection E.
Using a straightedge, connect points A and E
with a line.
The line that contains points AE is parallel to
line XB
C
X
B
D
A
E