Download Lesson #11 - mvb-math

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Subject Class Calendar Spring 2009
________________________
Subject M$F
Date Day Lesson
Teachers
Lesson #1 AIM: What are ratios and proportions?
Lesson #2 AIM: How do we prove triangles similar?
Lesson #3 AIM: What are other methods for proving
triangles similar?
Lesson #4 AIM: How can we prove proportions involving
line segments?
Lesson #5 AIM: How can we prove that products of line
segments are equal?
Lesson #6 Aim: What are the properties of the centroid of
a triangle?
Lesson #7 Aim: What is the Right-Triangle Altitude
Theorem?
Lesson #8 Aim: How do we apply the Right-Triangle
Altitude Theorem?
Lesson #9 AIM: How do we write the equation of a circle?
1/30
W
1/31
2/1
2/4
Th
F
M
2/5
T
2/6
W
2/7
Th
2/8
2/11
F
M
2/12
T
2/13
W
2/14
Th
Lesson #10 AIM: How do we find a common solution to a
quadratic-linear system of equations graphically?
Lesson #11 Aim: What are the parts of a circle?
Lesson #12 AIM: What are the properties of the four
centers of a triangle?
Lesson #13 Aim: How do we prove arcs congruent?
Lesson #14 Aim: How do we prove chords congruent?
Lesson #15 Aim: What relationships exist if a diameter is
perpendicular to a chord?
Lesson #16 Aim: How do we measure an inscribed
angle?
Lesson #17 Aim: What relationships exist when tangents
to a circle are drawn?
Lesson #18 Aim: How do we measure an angle formed
by a tangent and a chord?
Test
Lesson #19 Aim: How do we measure angles formed by
two tangents, a tangent and a secant, or two secants to a
circle?
Lesson #20 Aim: How do we measure angles formed by
two chords intersecting within a circle?
Lesson #21 Aim: How do we apply angle measurement
theorems to circle problems?
Lesson #22 Aim: How do we apply angle measurement
theorems to more complex circle problems?
HW
Com
plete
2
2/15
F
2/25
M
2/262/27
2/28
2/29
3/3
T/W
Th
F
M
Test
Lesson # 23 Aim: How do we use similar triangles to find
the measure of segments of chords intersecting in a circle?
Lesson # 24 Aim: How do we use similar triangles to find
the measure of line segments formed by a tangent and
secant to circle?
Lesson #25 Aim: How do we find the measures of secants
and their external segments drawn to a circle?
Review
Test 2
Lesson #26 Aim: How do we apply segment measurement
relationships to problems involving circles?
__________________
Subject M$G
Teachers Goldberg
Lesson #27 AIM: How do we determine a probable
locus?
Lesson #28 AIM: How do we solve problems using
compound loci?
Lesson #29 AIM: How do we find the equation of the
locus of points at a given distance from a given point?
Lesson #30 AIM: How do we write linear equations that
satisfy given locus conditions?
Lesson #31 AIM: How do we find the points in the
coordinate plane which satisfy two different conditions?
Lesson #32 AIM: How are images and pre-images
related under line reflections?
Lesson #33 AIM: How are images and pre-images
related under point reflections and translations?
Mar 20 Mon Lesson #34 AIM: How are images and pre-images
related under rotations?
Marc
Tues Lesson #35 AIM: How are images and pre-images
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related under dilations?
Lesson #36 AIM: How do we find an image under a
composition of transformations?
Lesson #37 AIM: Which transformations are
isometries?
Lesson #38 Aim: How do we apply the properties of
transformations to geometric proofs?
Lesson #39 Aim: What is solid geometry?
Lesson #40 Aim: How do we determine a plane?
Lesson #41 Aim: When is a line perpendicular to a
plane?
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Lesson #42 Aim: When are planes perpendicular?
Lesson #43 Aim: When are planes parallel?
Lesson #44 Aim: How do we find the volume and
surface area of prisms and cylinders?
Lesson #45 Aim: How do we find the volume and
surface area of pyramids and cones?
Lesson #46 Aim: What are the properties of a sphere?
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LESSON #1 AIM: WHAT ARE RATIOS AND PROPORTIONS?
Students will be able to
1.Define ratio, proportion, means, extremes, mean proportional, constant of
proportionality, alternation and inversion.
2. State and apply the theorem "In a proportion, the product of the means equals the
product of the extremes."
3. Determine if a proportion is valid.
4. Find the missing term of a proportion.
5. Find the mean proportional between two values.
6. Arrange four elements to form a valid proportion.
7. Form equivalent proportions using addition.
Do Now: Simplify in simplest form
Homework: Page 479#3-6 and #9,10,12,13,16
Vocabulary:
What is a ratio?
A ratio is a comparison of 2 numbers which can be written as follows:
a/b; a:b a to b
What is a proportion?
A proportion is an equation that states that 2 ratios are equal.
A:b=c:d then a and d are the extremes and b and c are the means.
Theorems:
 In a proportion the product of the means = the product of the extremes.
Example 3/2=12/8 2*12=3*8
 In a proportion the means or the extremes can be interchanged.
Example 3 = 12 or 8 =12 or 3 =2
2 8
2 3
12 8
Have students come up of one example of their own.
Mean Proportional
 If the 2 means of a proportion are equal either mean is called the mean proportional
between the extremes of the proportion.
Example
2/6-6/18 then 6 is the mean proportional also called the geometric mean.
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Find the mean proportional between 8 and 12.
8 x 2

x =96 x= 4 6

x 12
27 =9 9x+9==54 9x=45 x=5
x+1 2
Find the mean proportional between 9 and 8
9 =x x2=72 x=6
x 8
7 1
1
1
 and  b then  ?
10 a
a
b
7a= 10 then a =10/7 then b =7/10 and 1/b = 10/7 or 10b=7 and b=7/10 then 1/b =10/7
Advanced Algebra- SAT type questions
1. A gardener completes 5/8 of his landscaping jobs in 10 days. How many days will it
take him to finish the job?
.625 10
.625x=10 10*8/5=16 16-10=6

1
x
2. If 3 people work at the same rate and can paint a house in 8 days. What part of the
house can one person paint in 2 days?
It takes 24 man days for one person. 2 days would be 2/24 or 1/12
3. It takes 12 carpenters to build a house in 3 days, how many days will it take 8
carpenters?
12 8
12 x=24 x=2 days

3 x
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Lesson #2 AIM: How do we prove triangles similar?
Students will be able to:
1. create and state a definition for similar triangles and ratio of similtude
2. identify pairs of corresponding sides
3. compare and contrast the properties of triangles that are similar triangles and triangles
that are congruent
4. discover and apply the following similarity theorems to formal proofs:
a. If two triangles agree in two pairs of angles then these triangles are similar.
(2 Δs ~ by aa)
b. If the three sides of one triangle are proportional to the three corresponding sides of
another triangle, then the triangles are similar. (2 Δs ~ by SSS)
5. solve numerical and algebraic problems involving proportions in similar triangles.
Objects, such as these two cats, that have the same shape,
but do not have the same size, are said to be "similar".
The cat on the right is an
enlargement of the cat on the left.
They are exactly the same shape, but
they are NOT the same size.
These cats are similar figures.
The mathematical symbol used to denote
similar is .
Similar
Symbol
Do you remember this symbol as "part" of the
symbol for congruent??
Definition: In mathematics, polygons are similar if their corresponding (matching) angles are
equal and the ratio of their corresponding sides are in proportion. This definition allows for
congruent figures to also be "similar", where the ratio of the corresponding sides is
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Facts about similar triangles:
The ratio of the corresponding
sides is called the ratio of
similitude or scale factor.
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Lesson #3 AIM: What are other methods for proving triangles similar?
Students will be able to:
1. discover and apply the similarity theorems to formal proofs:
a. If two sides of one triangle are proportional to two sides of another triangle and their
included angles are congruent, then the triangles are similar. (2 Δs ~ by SAS)
b. If a line is parallel to one side of a triangle and intersects the other two sides, then it cuts
off a triangle similar to the original triangle.
2. state and prove: “If one or more lines are parallel to one side of a triangle and intersect
the other two sides, then the lines divide the two sides of the
triangle proportionally.
Two triangles are similar if and only if the corresponding sides are in
proportion and the corresponding angles are congruent.
There are three accepted methods of proving triangles similar:
To show two triangles are similar, it is sufficient to show that two angles of
AA
one triangle are congruent (equal) to two angles of the other triangle.
Definition:
Theorem: If two angles of one triangle are congruent to two angles of another triangle,
the triangles are similar.
SSS
for similarity
BE CAREFUL!! SSS for similar triangles is NOT the same theorem as
we used for congruent triangles. To show triangles are similar, it is
sufficient to show that the three sets of corresponding sides are in
proportion.
Theorem: If the three sets of corresponding sides of two triangles are in proportion, the
triangles are similar.
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SAS
for
similarity
BE CAREFUL!! SAS for similar triangles is NOT the same theorem
as we used for congruent triangles. To show triangles are similar, it is
sufficient to show that two sets of corresponding sides are in
proportion and the angles they include are congruent.
Theorem: If an angle of one triangle is congruent to the corresponding angle of another
triangle and the lengths of the sides including these angles are in proportion,
the triangles are similar.
Once the triangles are similar:
Theorem:
The corresponding sides of similar triangles are in proportion.
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Lesson #4 AIM: How can we prove proportions involving line segments?
Students will be able to:
1. prove and apply the following theorems in formal proofs to show line segments are in
proportion:
a. If two triangles are similar, then their corresponding angles are congruent and their
corresponding sides are in proportion.
b. If a line is parallel to one side of a triangle and intersects the other two sides, then the
line divides the two sides proportionally.
c. If a line segment joins the midpoints of two sides of a triangle, then it is parallel to the
third side and has length equal to one-half the length of the third side.
2. identify the triangles needed to be proven similar from a given proportion
3. write a proportion involving the corresponding sides of similar triangles
4. solve numerical and algebraic problems involving proportions in similar triangles
5. write proofs involving line segments that are in proportion
6. write proofs involving line segments that have a mean proportional
Lesson #5 AIM: How can we prove that products of line segments are equal?
Students will be able to:
1. Create a proportion from a given product of line segments
2. Identify the triangles needed to be proved similar
3. Prove, both formally and informally, triangles similar and line segments in proportion
4. Apply the theorem: "In a proportion, the product of the means equals the product of the
extremes." to prove products of lengths of line segments equal
Lesson #6
Aim: What are the properties of the centroid of a triangle?
Students will be able to:
1. state the definition of a median of a triangle
2. define centroid and concurrence
3. investigate the 2:1 relationship between the segments on the median formed by the
position of the centroid
4. locate the centroid of a triangle using measurement, construction, or manipulative tools,
such as paper folding, or dynamic geometry software
5. apply properties of the centroid to in-context situations
Lesson #7 Aim: What is the Right-Triangle Altitude Theorem?
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Students will be able to:
1. Define projection, mean proportional, geometric mean
2. Identify the altitude, hypotenuse, and projection on the hypotenuse given a diagram
3. Investigate, discover, and conjecture the right-triangle altitude theorem
a. Each leg of a right triangle is the mean proportional between its projection on the
hypotenuse and the whole hypotenuse.
b. The altitude drawn to the hypotenuse of a right triangle is the mean proportional
between the segments of the hypotenuse.
4. Express in writing the relationships between the measures of the segments involved in
the right-triangle altitude theorem in different contexts
Lesson #8 Aim: How do we apply the Right-Triangle Altitude Theorem?
Students will be able to:
1. state the Right-Triangle Altitude Theorem
2. apply the Right-Triangle Altitude Theorem to in-context numerical and algebraic
problems
3. apply the Right-Triangle Altitude Theorem in proofs
4. solve numerical problems related to similar triangles within a right triangle
"Mean Proportional" may also be referred to as a "Geometric Mean".
Remember the rule for working with proportions: the product of the means equals the
product of the extremes.
In a mean proportional problem, the "means" are the same values.
The mean proportional of two Positive numbers a and b is
the positive number x such that
. When solving,
Notice that the x value appears TWICE in the "means" Positions.
Theorem: The altitude to the hypotenuse of a right triangle forms two triangles that are
similar to each other and to the original triangle.
Since these triangles are similar, we can establish
proportions relating the corresponding sides. Two
valuable theorems can be formed using these
proportions.
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Theorem: The altitude to the hypotenuse of a right triangle is the mean proportional
between the segments into which it divides the hypotenuse.
Altitude Rule:
Theorem: Each leg of a right triangle is the mean proportional between the
hypotenuse and the projection of the leg on the hypotenuse.
Leg Rule:
or
Examples:
1. Find x:
Solution:
Examine the diagram to see what is given.
This problem needs the Altitude Rule.
4x = 64
x = 16
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2. Find x:
Solution:
Examine the diagram to see what is given.
This problem needs the Leg Rule (there is no
value on the altitude).
x2
= 16
x=4
(lengths are positive)
3. Find x to nearest tenth:
Solution:
Examine the diagram to see
what is given.
This problem needs the Leg
Rule (there is no value on the
altitude). But be careful in
this problem. You will need
the ENTIRE hypotenuse
length which is 4 + 12 = 16.
x2 = 192
x = 13.9
(lengths are positive)
a starfish, the outer leg length, AB, is the mean proportional between the length of the base of the leg,
4. In
BC, and the length of the outer leg plus the base, AC. The base of the leg is 4 cm. Find the length of the
outer leg. Use your graphing calculator to solve the quadratic equation, rounding your answer to the
nearest tenth.
Solution:
x2 = 4x + 16
x2 - 4x - 16 = 0
(USE GRAPHING
CALCULATOR TO
SOLVE. GRAPH AND
Use 2nd Calc #2 Zero to find the
positive root. x = 6.472136
Outer leg = 6.5 cm
FIND WHERE THE
GRAPH CROSSES THE XAXIS.)
WRITING EXERCISE: YOUR LESSON IN CLASS TODAY INTRODUCED YOU TO THE 'GEOMETRIC
MEAN.' USE THE INTERNET OR YOUR TEXTBOOK TO FIND OUT THE MEANING OF AN
'ARITHMETIC MEAN.' DESCRIBE THE DIFFERENCE BETWEEN THE ARITHMETIC AND GEOMETRIC
MEAN.
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Lesson #9 AIM: How do we write the equation of a circle?
Students will be able to:
1. use the distance formula to discover the equation of a circle with center at (h,k) and with
a radius of length r: (x−h)2 + (y−k)2 = r2
2. write the equation of a circle given any of the following:
a. the coordinates of the center and the length of the radius
b. the coordinates of the center and the coordinates of a point on the circle
c. the coordinates of the endpoints of a diameter
3. determine the coordinates of the center and the length of the radius of a circle whose
equation is given in center-radius form
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4. graph circles in the form (x−h)2 + (y−k)2 = r2
5. determine the center and radius of a circle whose graph is given (the coordinates of the
center and the length of the radius are integral values)
Lesson #10 AIM: How do we find a common solution to a quadratic-linear system of
equations graphically?
Students will be able to
1. graph the quadratic and linear equations on the same set of axes by making a table of
values, using a graphing calculator (include parabola and line as
well as circle and line), and/or using dynamic software
2. identify the coordinates of all common solutions by using calculator and/or dynamic
software intersection function
3. verify that the coordinates of each point of intersection is a common solution by
checking that they satisfy both equations
4. determine the number of solutions by inspecting the graph
Lesson #11
AIM: What are the parts of a circle?
Students will be able to
1. Define circle, radius, diameter, center, chord, secant, tangent, central angle, arc,
semicircle, minor arc, major arc, congruent arcs, and congruent circles
2. Discover, state, and apply the postulates:
a. In the same or congruent circles all radii are congruent
b. The degree measure of a central angle of a circle is equal to the degree measure of its
intercepted arc.
3. Explain the difference between arc degrees and arc length
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4. Solve numerical and algebraic problems involving diameters and radii, major and minor
arcs, and central angles
5. Apply the above definitions and postulates to formal and informal proofs
Do Now: Write and illustrate the definitions for the following terms.
Circle, radius, diameter, center of a circle.
Homework:
Page 440-441 # 4,5,6
Classwork: Page 440 # 3,
Definitions:
Circle- The set of all points in a plane such that the distance (radius) from a given point
(center of the circle) is constant.
Radius- Line segment from the center of the circle to any point of the circle.
Diameter- A chord that passes through the center of a circle
Center of Circle- The point in a circle from which the distance to any point on the circle is
a constant.
 All radii of the same circle are congruent.
Central Angle- An angle that has its vertex at the center of a circle.
Types of Arcs
 Arc- a continuous portion (as of a circle or ellipse) of a curved line
 Minor arc - < 180 degrees
 Major arc- > 180 degrees
 Semicircle -A 180 degree arc.
 Degree measure of an arc- equal to the measure of the central angle that intercepts it.
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 Congruent circle- circles with congruent radii.
 Congruent arcs- arcs in same or congruent circles that are equal.
All radii of the same circle are congruent.
Interior of a circle are all points less than the radius of a circle
Exterior of a circle are all points greater than the radius of a circle.
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 Arc Addition Postulate- Page 438
o If 2 arcs of the same circle having a common endpoint and no other points in
common then AB+BC=ABC
 In the same or in congruent circles, if two central angles are congruent, then the arcs
they intercept are congruent.
 In the same or in congruent circles, if two arcs are congruent, then their central angles
are congruent.
 The diameter of a circle divides the circle into two congruent arcs.
 Note that the angles and arcs must be in the same or in congruent circles
 Ask the students to draw 2 circles that are not congruent with equal central angles that
have equal arcs but whose arcs are not congruent.
Lesson #12 AIM: What are the properties of the four centers of a triangle?
Students will be able to:
1. define: cevian, concurrence, centroid, orthocenter, in-center, circumcenter
2. explain why medians, altitudes, angle bisectors, and perpendicular bisectors are cevians
3. investigate and locate the concurrency of cevians for a given triangle, using compass
and straight edge construction techniques
4. investigate relationships involving these four “centers”, such as the effects of the type of
triangle on the location of these centers, the location of the
centroid in relation to the length of each median, the conditions under which these four
centers are collinear, etc.
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5. discover the unique relationship the centroid and in-center have to the triangle, namely:
a. the centroid is the center of gravity
b. the in-center is the center of the inscribed circle
Lesson #13 AIM: How do we prove arcs congruent?
Students will be able to
1. state and apply the following:
a. The degree measure of a central angle of a circle is equal to the degree measure of its
intercepted arc.
b. In the same or in congruent circles, if two central angles are congruent, then the arcs
they intercept are congruent.
c. In the same or in congruent circles, if two arcs are congruent, then their central angles
are congruent.
d. In the same or in congruent circles, if two chords are congruent, then their arcs are
congruent.
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e. The diameter of a circle divides the circle into two congruent arcs.
2. apply the above theorems to numerical and algebraic problems, and to formal and
informal proofs
Do Now:
Homework: Page 440-441 # 6,7
 Chord - A segment that connects two distinct points on a circle
 Diameter- A chord that has one of its points the center of the circle.
 The measure of a central angle is equal to the measure of intercepted arc.
Theorem In the same or in congruent circles, congruent central angles have congruent chords
 In the same or in congruent circles, congruent arcs have congruent chords.
Lesson #14 AIM: How do we prove chords congruent?
Students will be able to
1. State that the distance from a point to a line is measured along the perpendicular from
the point to the line
2. Discover and apply the following:
a. In the same or in congruent circles, if two central angles are congruent, then the chords
they intercept are congruent.
b. In the same or in congruent circles, if two arcs are congruent, then their chords are
congruent.
c. In the same or in congruent circles, if two chords are equidistant from the center, then
they are congruent.
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3. Investigate the relative lengths of chords using their distance from the center of a circle
4. Apply the above theorems to numerical and algebraic problems and to formal and
informal proofs
Do Now:
Homework:
Theorem:
 In a circle, or congruent circles, congruent chords have congruent arcs.
 In a circle, or congruent circles, congruent arcs have congruent chords.
 In a circle or in congruent circles, congruent central angles have congruent chords.
 In a circle or in congruent chords, have congruent central angles.
Lesson #15 Chapter 11-2 AIM: What relationships exist if a diameter is perpendicular to
a chord?
Objectives: Students will be able to
 State and apply the theorem "If a diameter is perpendicular to a chord, then it
bisects the chord and its major and minor arcs."
 Apply the above to numerical and algebraic problems, and to Euclidean proofs.
Do Now:
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Given the labeled diagram at the left,
with diameter
. Find x. Find all arcs.
Homework: Page 447 # 6-11,15-18
Theorems:
 In a circle, a radius perpendicular to a chord bisects the chord. (See proof Page 443)
 In a circle, a radius that bisects a chord is perpendicular to the chord.
 In a circle, the perpendicular bisector of a chord passes through the center of the circle.
Theorem:
 In a circle, or congruent circles, congruent chords are equidistant from the center.
(See proof Page 444)
 In a circle, or congruent circles, chords equidistant from the center are congruent.
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Given: Circle O,
diameter, marked perpendicular
Find x.
Choose:
2.5
5
6.5
10
Given: Circle O,
AB = CD, marked perpendiculars
Find x.
Choose:
6
9
15
16
Given: Circle O,
Choose:
50
60
80
100
Given: Circle O,
Choose:
80
100
120
140
Given: Circle with indicated center,
marked parallels
Find x.
Choose:
40
80
120
240