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Transcript
```Geometry - Semester 2
Mrs. Day-Blattner
2/14/2017
Agenda 2/14/2017
1)
2)
3)
4)
5)
6)
7)
8)
Bulletin
Recap last time’s work
Circles, Chords, Diameters and their relationships - intro
Homework Hold Up and follow on activity
Exercises pages 18-21
Lesson Summary
Exit Ticket page 25
Homework
the point at which the
perpendicular bisectors of the
sides of a triangle intersect
and which is equidistant from
the three vertices
circumcenter
the point at which the
perpendicular bisectors of the
sides of a triangle intersect
and which is equidistant from
the three vertices
This circle circumscribes the
triangle - the 3 vertices lie on
the circumference
circumcenter
the point at which the
perpendicular bisectors of the
sides of a triangle intersect
and which is equidistant from
the three vertices
This circle circumscribes the
triangle - the 3 vertices lie on
the circumference
circumcenter
A_____B______C
if we have three points that
are colinear there could not
be a location equidistant
from all three points .
Lesson 2: Circles, Chords, Diameters, and their
Relationships (page 17)
Classwork
Opening Exercise
Construct the perpendicular bisector of the line
segment AB below.
A
B
Construct the perpendicular bisector of the line
segment AB below.
Draw another line that bisects segment AB but is not
perpendicular to it.
Draw another line that bisects segment AB but is not
perpendicular to it. List one similarity and one
difference between the two bisectors.
Draw another line that bisects segment AB but is not
perpendicular to it. List one similarity and one
difference between the two bisectors.
Similarity: Both lines
cut the segment into
two congruent parts cut it in half.
Difference:
Draw another line that bisects segment AB but is not
perpendicular to it. List one similarity and one
difference between the two bisectors.
Difference: All points
on the perpendicular
bisector are equal
distance from A and
B, but that is not true
for points on the
other line.
a point A is said to be
equidistant from two different
points B and C if AB = AC
equidistant
A
B
B
C
A
C
Hold up your homework - 3 different size circles with
2 congruent chords drawn in each one.
Now construct perpendicular bisectors for each
chord and extend the line so it goes through the
circumference of its circle twice.
What do we notice?
A
What do we notice?
The perpendicular bisector of a chord can generate
a diameter of the circle - it always goes through the
center of the circle.
Construct a circle of any radius and identify the
center as point P.
Draw a chord and label it AB.
What do you notice about the perpendicular bisector
of AB?
What can you say about the points on a circle in
relation to the center of the circle?
What can you say about the points on a circle in
relation to the center of the circle?
The center of the circle is equidistant from any two
points on the circle.
Look at the circles, chords, and perpendicular
bisectors created by your neighbors. What
statement can you make about the perpendicular
bisector of any chord of a circle? Why?
Look at the circles, chords, and perpendicular
bisectors created by your neighbors. What
statement can you make about the perpendicular
bisector of any chord of a circle? Why?
The perpendicular bisector of any chord must
always contain the center of the circle. The center of
the circle is equidistant from the two end points of
the chord because they lie on the circle. Therefore,
the center lies on the perpendicular bisector of the
chord. That is, the perpendicular bisector contains
the center.
How does this relate to the definition of the
perpendicular bisector of a line segment?
The set of all points equidistant from two given
points (endpoints of a line segment) is precisely the
set of all points on the perpendicular bisector of the
line segment.
Exercise Page 18 1st one together.
1. Prove the theorem: If a diameter of a circle
bisects a chord, then it must be perpendicular to
the chord.
Given: Circle C with diameter DE, chord AB, and
AF = BF
Prove: DE is perpendicular to AB
Proof version 1.
Statements
1.
2.
3.
4.
5.
6.
7.
AF = BF
FC = FC
AC = BC
triangle AFC is congruent to triangle
BFC
measure of angle AFC is equal to
measure of angle BFC
angles AFC and BFC are right
angles
Line segment DE is perpendicular
to line segment AB
Reasons.
1.
2.
3.
4.
5.
6.
7.
Given
Reflexive property
radii of same circle are equal in
measure
Side-side-side congruency
postulate
corresponding angles of congruent
triangles are equal in measure
equal angles that form a linear pair
each measure 90 degrees
Definition of perpendicular lines
Proof version 2.
Statements
1.
2.
3.
4.
5.
6.
7.
AF = BF
AC = BC
measure of angle FAC is equal to
measure of angle FBC
triangles AFC and BFC are
congruent
measure angle AFC = measure of
angle BFC
angles AFC and BFC are right
angles
Line segment DE is perpendicular
to line segment AB
Reasons.
1.
2.
3.
4.
5.
6.
7.
Given
radii of same circle are equal in
measure
base angles of an isosceles triangle
are equal in measure
SAS
Corresponding angles of congruent
triangles are equal in measure
equal angles that form a linear pair
each measure 90 degrees
Definition of perpendicular lines
2. Prove the theorem: If a diameter of a circle is
perpendicular to a chord, then it must bisect the
chord.
Given: Circle C with diameter DE, chord AB, and
DE is perpendicular to AB
Prove: DE bisects AB
2.Proof .
Statements
1.
2.
3.
4.
5.
6.
7.
8.
9.
Line segment DE is perpendicular to
line segment AB
angles AFC and BFC are right angles
angle AFC is congruent to angle BFC
AC = BC
measure of angle FAC is equal to
measure of angle FBC
measure of angle ACF is equal to
measure of angle BCF
triangles AFC and BFC are congruent
AF = BF
Line segment DE bisects line segment
AB
Reasons.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Given
Definition of perpendicular lines
all right angles are congruent
radii of the same circle are equal in
measure
base angles of isosceles triangles
are congruent
two angles of triangle are equal in
measure, so third angles are equal
ASA
corresponding sides of congruent
triangles are equal in length
Definition of segment bisector.
3. The distance from the center of a circle to a chord
is defined as the length of the perpendicular segment
from the center to the chord.
Note that since this perpendicular segment may be
extended to create a diameter of the circle , the
segment also bisects the chord.
A Central angle of a circle is an angle whose vertex
is the center of a circle.
3. Prove the theorem: In a circle, if two chords are
congruent, then the center is equidistant from the
two chords. Group 1
4. Prove the theorem: In a circle, if the center is
equidistant from the two chords, then the two chords
are congruent. Group 2
5. Group 3
6. Group 4
Lesson Summary - Read aloud and highlight
Theorems about chords and diameters in a circle
and their converses:
● If a diameter of a circle bisects a chord, then it
must be perpendicular to the chord.
● If a diameter of a circle is perpendicular to a
chord, then it bisects the chord.
Lesson Summary cont.
● If two chords are congruent, then the center is
equidistant from the two chords.
● If the center is equidistant from two chords, then
the two chords are congruent.
Lesson Summary cont.
● Congruent chords define central angles equal in
measure.
● If two chords define central angles equal in
measure, then they are congruent.
Lesson 2. Circles, Chords, Diameters and Their
Relationships. (page 25)
● Find the theorem from the lesson summary on
page 22 that justifies how you solve 1 and 2a) and
write it out in the space for solving that problem.
Then solve it!
● For 2b the circle still has a center P and a radius
of 10 and AB is still perpendicular to DE, BUT you
do not know the length of AB this time, you are
going to calculate what it will be with the new data.
Homework
Problem Set (starting on page 22)
Mandatory: 1 - 5.
Go ahead and solve the other problems
if you have time and interest.
```