Download The Perpendicular Bisector of a Segment

M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
The Perpendicular Bisector of a Segment
Check off the steps as you complete them:
______ Open a Geometry window in Geogebra.
______ Use the segment tool of draw AB .
______ Under the point tool, select Midpoint of Center, and create the midpoint of AB . (This
should be point C.)
______ Select the perpendicular line tool, and click on point C and AB .
______ Use the point tool to make point D anywhere on the perpendicular
line. Your diagram should look something like the one at right.
______ Select Distance or Length (in the Angle menu), and measure
the distances from D to A and from D to B by clicking on the
points.
______ Drag point D anywhere along the line, and observe the
measurements DA and DB.
1. What do you notice about DA and DB?
2. DC is called the perpendicular bisector of AB because it is perpendicular to AB and
passes through the midpoint of AB .
Complete this statement:
Any point on the perpendicular bisector of a segment is:
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
Follow the directions to find the equation of the perpendicular bisector of the line segment
through the given points.
3. Points:  2, 6  and  2, 4 
a. Graph the segment with endpoints
 2, 6  and  2, 4  .
b. Find the midpoint of the segment.
c. Find the slope of the segment.
d. Find the slope of a line perpendicular to this segment.
e. Use slope-intercept form y  mx  b or point-slope form y  k  m  x  h  , as appropriate
to find the equation of a line with slope from part (d) and point from part (b).
f. Graph the perpendicular bisector.
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
Follow the directions to find the equation of the perpendicular bisector of the line segment
through the given points.
4. Points: 1, 6  and  4,3
a. Graph the segment with endpoints
1, 6  and  4,3 .
b. Find the midpoint of the segment.
c. Find the slope of the segment.
d. Find the slope of a line perpendicular to this segment.
e. Use slope-intercept form y  mx  b or point-slope form y  k  m  x  h  , as appropriate
to find the equation of a line with slope from part (d) and point from part (b).
f. Graph the perpendicular bisector.
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
ASSIGNMENT 2M
Follow the directions to find the equation of the perpendicular bisector of the line segment through the given
points.
1. Points:  4,7  and  1, 3
a. Graph the segment with endpoints  4,7  and
 1, 3 .
b. Find the midpoint of the segment.
c. Find the slope of the segment.
d. Find the slope of a line perpendicular to this segment.
e. Use slope-intercept form y  mx  b or point-slope form y  k  m  x  h  , as appropriate to find the
equation of the perpendicular bisector.
f. Graph the perpendicular bisector.
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
Follow the directions to find the equation of the perpendicular bisector of the line segment through the given
points.
2. Points:  2,5  and  4,1
a. Graph the segment with endpoints  2,5  and
 4,1 .
b. Find the midpoint of the segment.
c. Find the slope of the segment.
d. Find the slope of a line perpendicular to this segment.
e. Use slope-intercept form y  mx  b or point-slope form y  k  m  x  h  , as appropriate to find the
equation of the perpendicular bisector.
f. Graph the perpendicular bisector.
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
Practice for Section 3-6
Follow the directions for each:
1. Points:  4, 2  and  8, 4 
a. Graph the segment with endpoints  4, 2  and  8, 4  .
b. Find the midpoint of the segment.
c. Find the slope of the segment.
d. Find the slope of a line perpendicular to this segment.
e. Find the equation of the perpendicular bisector.
f. Graph the perpendicular bisector.
g. Use algebra to rewrite your equation from part (e) in slope-intercept form y  mx  b .
h. Use algebra to rewrite your equation from part (g) in standard form (x and y on the left side, no decimals
or fractions).
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
2. Points:  4, 6  and  6, 2 
a. Graph the segment with endpoints  4, 6  and  6, 2  .
b. Find the midpoint of the segment.
c. Find the slope of the segment.
d. Find the slope of a line perpendicular to this segment.
e. Find the equation of the perpendicular bisector.
f. Graph the perpendicular bisector.
g. Use algebra to rewrite your equation from part (e) in slope-intercept form y  mx  b .
h. Use algebra to rewrite your equation from part (g) in standard form (x and y on the left side, no decimals
or fractions).
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
Review of 3-3 to 3-6
Figures in this review may not be drawn to scale.
Multiple Choice:
____ 1. In the diagram at right, TY MV , and mHRY  100 . Which of the following statements does not
have to be true?
(A)
(B)
(C)
(D)
mFHM  100
mMHR  80
TRS and FHM are alternate exterior angles
YRS and VHR are alternate interior angles
____2. In the diagram at right, which statement is the reason that mPIR  mYMS ?
When two parallel lines are cut by a transversal, then…
(A)
(B)
(C)
(D)
Alternate exterior angles are congruent
Alternate interior angles are congruent
Consecutive interior angles are supplementary
Corresponding angles are congruent
_____3. In the diagram at right, 2  3 . Which of the following must be true?
(A)
(B)
(C)
(D)
r t
m8  m6
m4  m6
m5  m3
____4. In the diagram at right, m6  m7  180 . Which of the following does not have to be true?
(A)
(B)
(C)
(D)
m1  m4  180
m5  m4  180
r s
m2  m7
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
____5. Annie wanted to solve for x, so she set up the equation 5x  2  3x  12  180 . What is her reason?
If two parallel lines are cut by a transversal, then…
(A)
(B)
(C)
(D)
Linear pairs are supplementary
Corresponding angles are supplementary
Alternate interior anlges are congruent
Consecutive interior angles are supplementary
____6. To solve for x in the diagram at right, Ben used the equation 9 x  5  10 x  5 . The reason that justifies
Ben’s equation is:
If two parallel lines are cut by a transversal, then…
(A)
(B)
(C)
(D)
Alternate interior angles are congruent
Alternate exterior angles are congruent
Corresponding angles are congruent
Consecutive interior angles are supplementary
7. Complete the proof:
Given: m n , r s , m1  130
Prove: m3  50
Statements
Reasons
1.
1.
2.
2. Corresponding s are 
3. m2  130
3.  s have = measures
4. m2  m3  180
4.
5. 130  m3  180
5.
6.
6.
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
8. Complete the proof:
Given: 1  2 , m3  88
Prove: m4  92
Statements
Reasons
1.
1.
2. c d
2.
3. m3  m4  180
3.
4.
4. Substitution
5.
5. Subtraction
Find the slope of each line.
9.
10.
11.
12.
Find the slopes of PQ and RS , and determine whether the lines are parallel, perpendicular, or neither.
13. P 1,0  , Q  5,3 , R  6, 1 , S  0,2 
14. P  5,1 , Q  1, 1 , R  2,1 , S  3, 2 
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
Graph each line, and write its equation.
15. the horizontal line through  3,1
16. the line with slope
Equation:
8
passing through 1, 5
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Equation:
17. the line through  1, 3 and  2, 4 
18. the line with x-intercept 2 and y-intercept 1
Equation:
Equation:
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
Graph each line.
19. y  3 
3
 x  1
4
20. y 
4
x2
3
Find the slope of each line.
21. y 
x
1
4
23. y  4 
22. 3x  2 y  6
24. (a) Graph MN with M  2,1 and N  2, 4  .
(b) Graph line
parallel to MN with y-intercept 4 .
(c) Write an equation for line
.
(d) Graph line k perpendicular to MN and passing through
1, 2  .
(e) Write an equation for line k.
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2
 x  2
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
25. Write an equation of the line through the point  4,5 and parallel to the line 2 x  y  6 .
26. Write an equation of the line through the point  3, 1 and perpendicular to x  3 y  2 .
27. (a) Graph AB with A 1, 5 and B  3,5 .
(b) Find the midpoint of AB .
(c) Find the slope of AB .
(d) Find the slope of a line perpendicular to AB .
(e) Find an equation of the perpendicular bisector of AB in point-slope form.
(f) Use algebra to rewrite your equation from (e) in slope-intercept form.
(g) Use algebra to rewrite your equation from (g) in standard form.
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
28. (a) Graph AB with A  5, 2  and B  1, 4  .
(b) Find the midpoint of AB .
(c) Find the slope of AB .
(d) Find the slope of a line perpendicular to AB .
(e) Find an equation of the perpendicular bisector of AB in point-slope form.
(f) Use algebra to rewrite your equation from (e) in slope-intercept form.
(g) Use algebra to rewrite your equation from (g) in standard form.
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
29. (a) Graph AB with A  6,0  and B  4, 5 .
(b) Find the midpoint of AB .
(c) Find the slope of AB .
(d) Find the slope of a line perpendicular to AB .
(e) Find an equation of the perpendicular bisector of AB in point-slope form.
(f) Use algebra to rewrite your equation from (e) in slope-intercept form.
(g) Use algebra to rewrite your equation from (g) in standard form.
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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW
30. (a) Graph AB with A  7,1 and B  3,5 .
(b) Find the midpoint of AB .
(c) Find the slope of AB .
(d) Find the slope of a line perpendicular to AB .
(e) Find an equation of the perpendicular bisector of AB in point-slope form.
(f) Use algebra to rewrite your equation from (e) in slope-intercept form.
(g) Use algebra to rewrite your equation from (g) in standard form.
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