Download 6.1 Warm Up The diagram includes a pair of congruent triangles

6.1 Warm Up
The diagram includes a pair of congruent triangles. Use the
congruent triangles to find the value of x in the diagram.
1.
2.
Write a proof.
3. Given: P is the midpoint of 𝑴𝑵 and 𝑻𝑸.
Prove: ∆𝑴𝑸𝑷 ≅ ∆𝑵𝑻𝑷
Geometry
6.1 Perpendicular and Angle Bisectors
[email protected]
6.1 Essential Question
What conjectures can you make about a point on
the perpendicular bisector of a segment and a
point on the bisector of an angle?
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
3
Goals



Know what a perpendicular bisector is.
Use the Perpendicular Bisector Theorem.
Measure the distance to a line.
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
4
6.1 Perpendicular and angle bisectors
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
5
The perpendicular bisector of a segment
can be a segment.
R
A
B
S
RS is a  bisector of AB.
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
6
The perpendicular bisector of a segment
can be a line.
R
A
B
S
RS is a  bisector of AB.
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
7
The perpendicular bisector of a segment
can be a ray.
R
A
B
S
RS is a  bisector of AB.
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
8
The perpendicular bisector of a segment
can be a plane.
K
A
B
K is a  bisector of AB.
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
9
Equidistant Points

A point is equidistant from two points if its
distance to each point is the same.
R
A
B
S


R is equidistant from A and B.
S is also equidistant from A and B.
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
10
How to use Geogebra
The tool bar at the top will have various tools listed.
Today’s exploration has the following tool bar…
This tool allows you to drag or select objects.
This tool allows you to find the distance between two
points or the length of a segment.
This allows you to move the entire picture.
Use these to zoom in or out as needed.
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
11
Exploration 1
Work with a partner. Open the 6.1 Exploration 1 from the ebook.
Given 𝐴𝐵 with point C on the perpendicular bisector of 𝐴𝐵.
Use the tool box to measure AC and BC.
Move points A, B, and C around.
Q1: What do you notice about
the distances from C to both
A and B?
Q2: What can you say about
the distances from any point
on the perpendicular bisector
of a segment to the endpoints
of the segment?
Theorem 6.1 (Perpendicular Bisector Theorem)
AR  BR
If a point is on the
perpendicular
bisector of a
segment, then it is
equidistant from the
endpoints of the
segment.
R
A
B
S
AS  BS
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
13
Theorem 6.1 Proof
1.
2.
The two right
triangles are
congruent by SAS.
This means AR = BR
by CPCTC.
January 13, 2016
R
A
Geometry 5.1 Perpendiculars and Bisectors
B
14
Theorem 6.2 (Converse of Perpendicular Bisector Thm.)
If a point is
equidistant from the
endpoints of a
segment, then it is on
the perpendicular
bisector of the
segment.
January 13, 2016
R
A
Geometry 5.1 Perpendiculars and Bisectors
B
15
Example 1
Find RB.
R
14
14
?
A
January 13, 2016
B
Geometry 5.1 Perpendiculars and Bisectors
16
Example 2
R
7.8
8
A
S
B
Is RS the perpendicular bisector of AB?
No, RA  RB.
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
17
Example 3
Find AD.
From the figure, 𝐵𝐷 is the
perpendicular bisector of 𝐴𝐶.
AD = CD Perpendicular Bisector Theorem
5x = 3x + 14 Substitute.
x = 7 Solve for x.
So, AD = 5x = 5(7)
AD = 35
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
18
Distance from a point to a line.
Defined as the length of the perpendicular
segment between the point and the line.
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
19
Distance from a point to a line.
Defined as the length of the perpendicular
segment between the point and the line.
This is the distance from
the point to the line.
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
20
Equidistant from two lines.
Point J is equidistant from lines m and n.
m
J
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
n
21
Exploration 2
Work with a partner. Open the 6.1 Exploration 2 from the ebook.
Given ∠𝐵𝐴𝐶 with point D on the bisector of ∠𝐵𝐴𝐶.
Use the tool box to measure DE and DF.
Move points B, C and D around.
Q1: What do you notice about
the distances from D to both
E and F?
Q2: What can you say about
the distances from any point
on the angle bisector to the
sides of the angles?
Q3: Why did we used 𝐷𝐸 and 𝐷𝐹 instead of using 𝐷𝐵 and 𝐷𝐶?
Theorem 6.3 (Angle Bisector Theorem)
If a point is on the bisector of an angle, then it
is equidistant from the two sides of the angle.
A
D
C
B
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
23
Theorem 6.4 (Converse of Angle Bisector Theorem)
If a point is equidistant from the two sides of the angle,
then it is on the bisector of an angle.
A
D
C
B
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
24
Example 4
15
If AD = 15, then DC =____.
A
D
C
B
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
25
Example 5
Is D on the bisector of the angle?Yes
4
D
4
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
26
Example 6
Is E on the bisector of the angle? NO!
The length of the
segment from E
to each is not the
length of a
perpendicular
segment.
January 13, 2016
5
E
5
Geometry 5.1 Perpendiculars and Bisectors
27
Example 7
𝑩𝑫 bisects ∠ABC, AD = 3z + 7,
and CD = 2z + 11. Find CD.
AD = CD
Angle Bisector Theorem
3z + 7 = 2z + 11
Substitute.
z=4
Solve for z.
So, CD = 2z + 11 = 2(4) + 11
CD = 19
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
28
Summary

A point on the
perpendicular bisector
of a segment is
equidistant from the
endpoints of the
segment.
AR  BR
R
A
B
S
AS  BS
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
29
Summary

A point on the bisector
of an angle is
equidistant from the
sides of the angle.
A
D
C
B
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
30
Summary

The distance from a point to a line is the
measure of the perpendicular segment.
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
31
Homework
January 13, 2016
Geometry 5.1 Perpendiculars and Bisectors
32