Geometry
Construction #1
Construct a segment congruent to a given segment.
This is our compass.
Given:
Procedure:
A
B
1. Use a straightedge to draw a line.
Call it l.
Construct: XY = AB
2. Choose
point on l and label it X.
Don’t change
yourany
radius!
l
X
Y
3. Set your compass for radius AB and
make a mark on the line where B lies.
Then, move your compass to line l and
set your pointer on X. Make a mark on
the line and label it Y.
Construction #2
Construct an angle congruent to a given angle
A
Given:
Procedure:
D
1) Draw a ray. Label it RY.
B
E
C
Construct:
D2
R
E2
Y
2) Using B as center and any radius,
draw an arc intersecting BA and
BC. Label the points of intersection
D and E.
3) Using R as center and the SAME
RADIUS as in Step 2, draw an arc
intersecting RY. Label point E2 the
point where the arc intersects RY
4) Measure the arc from D to E.
5) Move the pointer to E2 and
make an arc that that intersects the
blue arc to get point D2
6) Draw a ray from R through D2
Construction #3
How do I construct a
Bisector of a given angle?
A
Given:
Z
X
B
Procedure:
1.
Y
C
Using B as center and any radius, draw
and arc that intersects BA at X and BC at point Y.
2. Using X as center and a suitable radius, draw and arc.
Using Y as center and the same radius, draw an arc
that intersects the arc with center X at point Z.
3. Draw BZ.
Construction #4
How do I construct a perpendicular
bisector to a given segment?
X
Given:
B
A
Y
Procedure:
1.
Using any radius greater than 1/2 AB, draw four arcs of
equal radii, two with center A and two with center B.
Label the points of intersection X and Y.
2.
Draw XY
Construction #5
How do I construct a perpendicular bisector
to a given segment at a given point?
Z
C
Given:
Procedure:
X
1.
2.
3.
k
Y
Using C as center and any radius, draw arcs intersecting
k at X and Y.
Using X as center and any radius greater than CX,
draw an arc. Using Y as center and the same radius,
draw and arc intersecting the arc with center X at Z.
Draw CZ.
Construction #6
How do I construct a perpendicular bisector to a
given segment at a given point outside the line?
P
Given:
k
X
Y
Z
Procedure:
1.
2.
3.
Using P as center, draw two arcs of equal radii that
intersect k at points X and Y.
Using X and Y as centers and a suitable radius, draw arcs
that intersect at a point Z.
Draw PZ.
Construction #7
How do I construct a line parallel to a
given line through a given point?
P 1
Given:
Procedure:
l
k
A
B
1.
Let A and B be two points on line k. Draw PA.
2.
At P, construct <1 so that <1 and <PAB are congruent
corresponding angles. Let l be the line containing the
ray you just constructed.
Concurrent Lines
If the lines are Concurrent then they all intersect at the
same point.
The point of intersection is called the “point of concurrency”
What are the 4 different types of concurrent lines for a triangle?
Concurrent Lines
Point of Concurrency
I.
Perpendicular bisectors
Circumcenter
II.
Angle bisectors
Incenter
III. Altitudes
Orthocenter
IV.
Centriod
Medians
Circumcenter
Incenter
SP
Orthocenter
SP
SP
Centroid
SP
Construction #8
Given a point on a circle,
construct the tangent to the circle at the given point .
Given: Point A on circle O.
5
PROCEDURE:
1) Draw Ray OA
2) Construct a perpendicular
through OA at point A.
3
X
3) Draw tangent line XY
2
1
O
A
P
6
Y
4
Q
Construct
& 4 using
Now, usingarcs
the3same
radius,
point
Q asarcs
the center
and any
construct
5 & 6 using
Construct
1 and
2that
suitable
radius
(keep
this
point
P as
thearcs
center
so
using
any suitable
radius
radius)
they
intersect
arcs 3 &
4 to
A asXthe
getand
points
& center
Y
Construction #9
Given a point outside a circle,
construct a tangent to the circle from the given point.
Given: point A not on circle O
PROCEDURE:
1) Draw OA.
X
3
1
3) Construct a 2nd circle with center
M and radius MA
A
4) So you get points of tangency at
X & Y where
the arcs
intersect
Construct
arcs
1&
2
using
Construct
arcs 3& 4 using
the red circle
a suitable
radius
greater
5) the
Draw
tangents
AX & AY
same
radius
M
O
Y
4
2) Find the midpoint M of OA
(perpendicular bisector of OA)
2
than ½AO
(greater than ½AO)
( keep
Youthis
getradius
arcs 5for
& 6the
next step)
Construction #10
Given a triangle construct the circumscribed circle.
Given: Triangle ABC
B
3
6
1
F
A
8
5
4
PROCEDURE:
7
2
1) Construct the perpendicular
bisectors of the sides of the triangle
and label the point of intersection F.
Bisect
From point
segment
B construct
BC; Using
2) Set your compass pointer
ato
arcs
radius
7 F&and
greater
8 the
andradius
draw
thanto a line
point
measure
FC. the
1/2BC
connecting
from
point C
3)
Draw the arcs
circleof
construct
intersections
5with
&
thecenter
6 arcsF ,
C that passes through the vertices
A, B, & C
Now construct
the perpendicular
bisector
of segment
and a3label
Bisectpoint
segment
AC;AB
Using
From
A construct
arcs
&
where
theconnecting
31/2AC
lines meet.
radius
greater
than
from
4point
and F,
draw
a line
the
point C construct
intersections
of thearcs
arcs1 & 2
Construction #11
Given a triangle construct the inscribed circle.
PROCEDURE:
Given: Triangle ABC
1) Construct the angle bisectors of
angles A, B, & C, to get a point of
intersection and call it F
B
2) Construct a perpendicular to
side AC from point F, and label
this point G.
F
A
X
G
3) Put your pointer on point F and
set your radius to FG.
C
Y
4) Draw the circle using F as the
center and it should be tangent to all
the sides of the triangle.
Construction #12
Given
a segment,
Remember
you made
3
youofare
dividing parts.
divide the segment into a givenbecause
number
congruent
by 3, but if you wanted to
divide by,
say, 6 you
Given: Segment AB
PROCEDURE:
Divide AB into 3 congruent parts.would have to make 6
congruent
parts
on RAY
the from
1) Construct
ANY
raypoint
and A
sothat’s
on for
not7,8,9…
AB
A
C
D
B
2) Construct 3 congruent
segments on the ray using ANY
RADIUS starting from point A.
Label the new points X, Y, & Z
X
Y
So AC=CD=DB
Use any suitable radius that
Three
congruent
will give some
distance
keep the
betweenRemember
the pointssegments
same radius!!
1
Z
3) Draw segment ZB and copy the
angle AZB ( 1) to vertices X & Y
4) Draw the the rays from X & Y,
they should be parallel to the
segment ZB and divide AB into 3
congruent parts.
Construction #13
Given three segments construct a fourth segment (x) so
that the four segments are in proportion.
a
Given:
Construct: segment x such that
a
b
c
x
b
c
b
x
PROCEDURE:
a
1) Using your straight edge construct
an acute angle of any measure.
1
c
2) On the lower ray construct “a”
and then “c “ from the end of “a”.
3) On the upper ray construct “b” and
then connect the ends of “a & b”
4) Next copy angle 1 at the end of “c”
and then construct the parallel line
Construction #14
Given 2 segments construct their geometric mean.
Mark off 2 arcs from Y.
a
Keep the same
radius
and that:
Given:
Construct:
segment
x such
sure
radius
b Make
mark
offto2set
more
arcs
from
a your
x toX,more
thancrossing
1/2NM the
then:
xfirst two. b
Mark
2 arcs
from M.on
Keep
the same
Mark
off off
2 equal
distances
either
radius
andOmark
2 more
arcs
orange
segment
is xoff
the
PROCEDURE:
of point
using
any
radius
andfrom N,
9 7 3 The
1 side
Q
K then
crossing
the
first
two.
geometric
mean
between
the
bisect
this
new
segment
1)
Draw
a ray and mark off a+b.
the bperpendicular
bisector
PQpoint M.
lengthsDraw
of a and
2) Bisect a+b (XY)
and label
6
5
through point
O from O to M,
X
Y 3) Construct
M (I used the distance
the circle with center M
N a O
but
b remember any
andradius
radius = will
MY do)
(or MX)
X
8 10 2 4
P
L
4) Construct a perpendicular where
segment “a” meets segment “b”
(point O)
The Meaning of Locus
If a figure is a locus then it is the set of all points that
satisfy one or more conditions.
The term “locus” is just a technical term meaning “a set of
points”.
So , a circle is a locus.
Why??
Because it is a set of points a given distance from a
given point.
What can a locus be??
Remember it is a SET OF POINTS so if you recall the idea
of sets from algebra it is possible for a set to be empty.
So a set could be:
A. The empty set. (no points fit the condition or conditions)
B. A single point.
C. Two points, three points….
D. An infinite set of points. (like a line, circle, curve,…)
Examples of a Locus in a Plane
Locus Problems