12-3: Other Constructions (pg. 780-785)
Properties of a Rhombus
1. A _________________________ is a parallelogram in which all sides are congruent.
2. A __________________________ in which all sides are congruent is a rhombus.
3. Each __________________ of a rhombus bisects the opposite angles.
4. The diagonals of a rhombus are _______________________.
5. The diagonals of a rhombus __________________ each other.
Word Bank:
Perpendicular
Diagonal
Bisect
Rhombus
Quadrilateral
Constructing a Parallel Line
Method 1: Rhombus Method
Objective: To construct a line
through P parallel to t
.
P
P will become a vertex of a
rhombus.
1.Through P, draw any line
that intersects t. Call it A. PA
will be a side of the rhombus.
2.Place the pointer of your
compass on point A and place
pencil of compass on point P.
Then draw an arc so that it
intersects line t. This will be
the third vertex, X.
3.With the same opening
length of the compass, draw
intersecting arcs- first with the
point at P, then second, with
the point at X. This will be
labeled Y, the fourth vertex.
4.Draw line segment PY. It is
parallel to t.
t
.
P
t
Now you try:
.
P
t
Method 2: Corresponding Angle Method
Objective: To construct a line
through P that is parallel to
line t.
.
P
t
1.Through P, draw a line that
intersects t. Label it point A.
2.Label the angle you created
as X. Extend line AP well
beyond point P.
3.Copy angle a at point P.
(Remember from 12-1: use
compass to measure line
segment AP. Make an arc
keeping point of compass on
point A. Then use that length
to make an arc with the point
placed on point P. Where that
arc intersects line AP, place
the compass and draw
another arc.)
4.The two arcs will intersect
indicating point Q. Then draw
a line through points P and Q.
This is your parallel line.
.
P
t
Constructing Angle Bisectors
An _____________ _____________ is a ray that separates an angle into two congruent angles.
To bisect a given angle, we
make <A an angle of a
rhombus.
A
1.With the pointer of the
compass at A, draw any arc
intersecting the two line
segments. Label them B and C.
(This will give three vertices of
the desired rhombus).
2. Measure AB with the
compass. Place the pointer at
B and draw and arc. Do the
same for AC and point C. The
arcs intersect at D (the fourth
vertex).
A
3. Connect A with D. Ray AD is
the angle bisector of <A.
Given Triangle ABC, find the center using angle bisectors.
A
B
C
Constructing Perpendicular Lines
Objective: to construct a
perpendicular to t through P
by making P a vertex of a
rhombus.
.
P
1.Draw an arc with center at P
that intersects the line at two
points, A and B (these will be
two more vertices of the
rhombus).
.
P
2. With the same compass
opening, make two
intersecting arcs, one with
center at A and the other with
center at B. The arcs intersect
at Q (the final vertex of the
rhombus).
3. Connect P with Q. PQ and
AB are perpendicular
diagonals in rhombus PAQB.
Objective: to construct a
perpendicular bisector of line
segment AB. A and B will be
the endpoints of a diagonal of
a rhombus.
1.Draw any two intersecting
arcs with the same radius (one
with center at A and the other
at B) to determine line
segment CD.
2. The line CD is perpendicular
to AB. Point M is the midpoint.
A
A
B
B
Objective: to construct a
perpendicular to t at point M.
M will be the point of
intersection of the diagonals
of a rhombus.
.
t
M
1.Draw any arc with center at
M that intersects line t at two
points, A and B. AB is the
diagonal of a rhombus.
2. Use a larger opening for the
compass and draw
intersecting arcs, with centers
at A and B. Name the points of
intersection of the larger arcs
C and D. This is the other
diagonal of the rhombus.
.
t
M
3. Connect C with D. CD is
perpendicular to AB.
Notice that each shape we have constructed is a rhombus with congruent sides. Its diagonals bisect
each other and are perpendicular.
An __________________ is the segment perpendicular from a vertex containing the opposite side of
a triangle.
Use the methods above for these examples: