Download Geometry - Semester 2

Geometry - Semester 2
Mrs. Day-Blattner
1/28/2016
Agenda 1/28/2016
1) Turn in Paris trip if finished it at home
2) Race to the Center of the Pool and Circles, Chords,
Diameters and their relationships
3) Homework
“My trip to Paris”
1. What influenced your choice of location for the
hotel?
2. What is the name for the location of the hotel?
3. Why do you think the instructions specified your
locations could not be in a line?
the point at which the
perpendicular bisectors of the
sides of a triangle intersect
and which is equidistant from
the three vertices
circumcenter
Learning Log (important things to remember)
We can only find the circumcenter of 3 points that are not
colinear - if we have three points that are co-linear there
could not be a location equidistant from all three points of
interest.
(We could find a point closer to one or 2, but not all 3.)
A.
B.
C.
Lesson 4: Central Angles and the chords they
subtend. Turn over to…
RACE TO THE CENTER OF THE POOL
You are about to make some predictions that will
reveal how much you already know about the
Geometry of circles and chords.
The diagram below represents a large circular
swimming pool.
In the middle of the pool, at a point E, is an elevated
lifeguard chair. Chords AB and CD represent
catwalks that stretch across the pool.
A
C
E.
D
B
Xena is standing on catwalk AB and Yuliza is on
catwalk CD. Xena and Yuliza are identical twins,
and they are equally fast swimmers in a race to the
lifeguard chair.
A
C
E.
D
B
5 minutes. You and your partner read questions and
answer them as best you can.
A
C
E.
B
D
(pg 19) Lesson 2: Circles, Chords, Diameters, and
their Relationships.
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.
a point A is said to be
equidistant from two different
points B and C if AB = AC
equidistant
On a piece of scratch paper draw 4 circles and then
draw a chord of each one. What do you notice?
On a piece of scratch paper draw 4 circles and then
draw a chord of each one. What do you notice?
Now draw the perpendicular bisectors for each
chord and extend the line so it goes through the
circumference of the circle twice.
On a piece of scratch paper draw 4 circles and then
draw a chord of each one. What do you notice?
Now draw the perpendicular bisectors for each
chord and extend the line so it goes through the
circumference of the circle twice.
The perpendicular bisector of a chord gives 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?
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?
It passes through point P, the center of the circle.
Draw another chord and label it CD.
Construct the perpendicular bisector of CD.
What do you notice about the perpendicular bisector
of CD?
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?
It pass through point P, the center of the circle.
Draw another chord and label it CD.
Construct the perpendicular bisector of CD.
What do you notice about the perpendicular bisector
of AB?
It passes through point P, the center of the circle.
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.
Exercises 1 - 6 (start on page 20)
1.
2.
Nick, Ervin, David, Rafi
Eleazar, Adrian, Evelyn, Rosa
3.
Lorenzo, Sedrik, Nerses, Peter
4.
5.
6.
Ani, Leo, Dahsan, Mihail
Leslie, Zeke, Harut, Leon
Edgar, Georgiy, Melanis, Preny
1.Catrina, Natalie, Aren, Tigran
2. Liam, Michelle, Jake, Kirsten
3. Karlen, Alec, Vahan, Jerry
2 column
proof with
fully labelled
diagram
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.
Lesson Summary
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.
Homework
Problem Set (starting on page 23)
1 - 8.
And complete Activity 4. Making and
Proving Conjectures about Congruent
Chords after clicking on the link below
http://projectsharetexas.org/resource/using-logical-reasoning-prove-conjectures-about-circles-ontrack-geometry-module-1lesson-19