Download 3x + 120 - x = x - 5x - Manhasset Public Schools

ANSWER KEY (arcs of circles) All construction arcs and lines are in bold.
Name ___________________
CC Geometry H
Date _____________________
HW #8
#1- 4 Bisect each angle below (angle that is marked with an arc). The arc
shown is not part of your construction.
2)
1)
4)
3)
5) Copy the angle below.
6) Construct and label BD, the bisector of
Construct
ABC.
XYZ, such that m ABD = m XYZ. X
D
C
Y
B
A
Z
7) Find the measures of the three interior angles of the triangle. Show all steps.
120 - 15 = 105
2
0
(120 ­ x)0
180 - 150 = 30
(3x)0
3(15) = 45
0
0
(x2 ­ 7x)0
2
3x + 120 - x = x - 5x
2
x - 7x - 120 = 0
(x - 15)(x + 8) = 0
x = 15 x= - 8 reject
(15) - 7(15) = 150
0
ANSWER KEY (Full circles) All construction circles and lines are in bold.
Name ___________________
CC Geometry H
Date _____________________
HW #8
#1- 4 Bisect each angle below (angle that is marked with an arc). The arc
2)
shown is not part of your construction.
1)
4)
3)
5) Copy the angle below.
6) Construct and label BD, the bisector of
Construct
ABC.
XYZ, such that m ABD = m XYZ. X
D
C Dashed circles
show the copying of
ABD. (second part B
A
Y
Z
of construction)
7) Find the measures of the three interior angles of the triangle. Show all steps.
2
0
120 - 15 = 105
3x + 120 - x = x - 5x
x2 - 7x - 120 = 0
(x - 15)(x + 8) = 0
reject
x = 15 x= - 8
(120 ­ x)0
(3x)0
3(15) = 45
0
180 - 150 = 30
0
(x2 ­ 5x)0
2
(15) - 7(15) = 150
0
Aim #9: How do we construct a perpendicular bisector?
CC Geometry H
Do Now:
Using the angle below:
1. Bisect the angle. Label the bisector BD.
2. Construct a copy of ≮DBC using vertex B'.
D
A
B
C
B'
Relevant Vocabulary
Two lines are PERPENDICULAR (
) if they intersect in one point, and any of the
angles formed by the intersection of the lines is a __________ angle.
(Two segments or rays are perpendicular if the lines containing them are
.)
The MIDPOINT of a segment divides a segment into 2 = or _____ parts.
A SEGMENT BISECTOR passes through the ____________ of a segment.
An ANGLE BISECTOR is a ray (line/segment) that divides an _____ into 2 = or ≅ parts.
AJ has been constructed as the angle
bisector of ≮BAD.
B
• Draw BD.
• Label C, the point of intersection of BD and AJ.
A
J
D
Notice:
(1) BC = CD so C is a ________________.
Therefore AJ is a ______________ of BD.
(2)≮BCA = ≮DCA. Each of these angles measures____.
Therefore, BD and AJ are _______________.
AJ is the ______________________ _________________________ of BD.
The perpendicular bisectorof a line segment passes through
the
________________of the segment and forms___________ angles with the
segment. (It is perpendicular to a segment at its midpoint.)
Using a compass and straightedge, we will now construct a perpendicular bisector
of a line segment. Experiment with your construction tools and the following
line segment to establish the steps that result in the perpendicular bisector.
[Use what you know about constructing an equilateral triangle.]
http://www.mathsisfun.com/geometry/construct­linebisect.html
A
B
Steps for Creating a Perpendicular Bisector.
1. Draw circle A: center A, radius > 1/2AB and circle B: center B, radius > 1/2BA.
2. Label the points of intersections as C and D.
3. Draw CD.
Construct the perpendicular bisector of CD.
• Label the midpoint M and labelthe perpendicular bisector as EF.
• Name one right angle:__________.
C
D
Relevant Vocabulary: EQUIDISTANT
A point A is said to be equidistant from two different points B and C if AB = AC.
Draw a diagram.
CDE is the perpendicular bisector of AB. (C, D, and E are collinear points.)
Using your compass, what conclusion can you make about the following
pairs of segments?
1) AC and BC
2) AD and BD
3) AE and BE
Based on your findings, fill in the observation below.
Any point on the perpendicular bisector of a line
segment is _____________________ from the
endpoints of the line segment.
Why?
Now construct a perpendicular line to line l from a point X not on line l .
The steps of the construction have been outlined below for you.
X
l Step 1: Draw circle X so that the circle intersects line l in two points.
Step 2: Label the two points of intersection as B and C.
Step 3: Draw circle B: center ____
, radius ______.
Step 4: Draw circle C: center ____, radius ______.
Step 5: Label the unlabeled intersection of circle B and circle C as D .
Step 6: Draw the perpendicular bisector: line _______.
Exercises
1. Divide segment AB into 4 segments of equal length.
B
A
2. Construct parallel lines l1 and
Step 1: Construct line
Step 2: Construct line
l2
as follows:
l3 which will be perpendicular to line l1 from point A
l2 which will be perpendicular to l3 through point A.
(Hint: This is the same as bisecting a straight angle.)
A
l1
3. Here is another method for constructing a line parallel to a given line through
a point not on the line, not using perpendicular lines.
Using the construction for copying an angle, construct a line parallel to line L
through point P.
P
L
4a) Construct the perpendicular bisector of BC.
b) Construct the angle bisector of ≮B.
C
A
B
Sum it Up!!
A perpendicular bisector of a segment passes
midpoint
through the ______________of
the segment
right
and forms ________ angles with the segment.
(Mark the diagram to show this.)
A
A point A is said to be equidistant from two
different points B and C if AB = AC.
(Mark the diagram to show this.)
C
B
point A would lie on the perp. bisector of BC!
Name ______________________
Date ________________
CC Geometry H
HW #9
1. Construct the perpendicular bisector of the segment below.
A
B
2. Construct the line perpendicular to line
l through point A.
List all the steps necessary to complete the construction.
A
l
3. Construct the perpendicular bisectors of AB, BC , and CA on the triangle below.
What do you notice about the segments you have constructed?
A
B
C
OVER
4. Two homes are built on a plot of land. Both homeowners have dogs, and are
interested in putting up as much fencing as possible between their homes on the
land, but in a way that keeps the fence equidistant from each home. Use your
construction tools to determine where the fence should go on the plot of land.
5. How will the fencing alter with the addition of a third home?
Review
2
1. In ΔABC, ≮B = ≮C. If AB = x + 8x, BC = 3x + 5 and AC = 20, find the
perimeter ΔABC.
2. In the diagram below, ≮BCA = 8x – 15, ≮DBC = 2x + 12 and ≮BDC = 4x + 9.
Find m≮BCD.
Construction of a perpendicular to a line from a point not on the line -
arcs (not full circles)
A
step 3
l step 1
step 2
step 2