Download two triangles are similar

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Chapter 11, Similarity
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Agenda:
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Review ratio and proportion
Cover Chapter 11 this week
Reminder: bring scientific calculator next time for Chapter 12
(trig)
A ratio is an expression that compares two quantities by
division, examples: a/b, a to b, a:b; we will use the fraction
form.
A proportion is a statement of equality between two ratios.
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Example A: A car travels 106 miles on 4 gallons of gas. How far
can it go on a full tank of 12 gallons?
Example B: Solve
x 1
4
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2 x  3 13
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11.1 Similar Polygons
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Objective:
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learn what it means for two figures to be similar
 Use the definition of similarity to find missing measures in similar polygons
 Explore the dilations of figures on a coordinate plane
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Two polygons are similar if and only if
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The corresponding angles are congruent
 The corresponding sides are proportional
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Examples on the board
A dilation is a non-rigid transformation of a figure which preserve the
shape, but either stretch the sides or shrink the sides with same proportion.
To dilate a figure in the coordinate plane, is to multiply the coordinates of
all its vertices by the same number, called scale factor.
Examples on the board
Dilation Similarity Conjecture: If one polygon is dilated image of another
polygon, then the polygons are: ___________________.
Homework # 69, pp. 585-587 # 1-2, 4-18 even
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11.2a Similar Triangles
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Objective: Learn shortcuts for determining whether two triangles are
similar
Remember the congruent shortcuts?
Investigation: If two angles of one triangle are congruent to another
triangle, must the two triangles be similar?
AA Similarity Conjecture: If _________ angles of one triangle are
congruent to _________ angles of another triangle, then the two
triangles are __________________. (like the congruent shortcut:
ASA, AAS)
Investigation: If three sides of one triangle are proportional to the
three sides of another triangle, must the two triangle be similar?
SSS Similarity Conjecture: If the three sides of one triangle are
proportional to the three sides of another triangle, then the two
triangles are ______________.
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11.2b Similar Triangles
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Investigation: Is SAS a similarity shortcut? What is SAS it means
corresponding sides are proportional and the angle between the sides are
congruent.
SAS Similarity Conjecture: If two sides of one triangle are proportional to
two sides of another triangle, and the included angles are congruent, then
the two triangles are similar.
Examples on the board
Is SSA a similarity shortcut? No, give a counter example.
If you know two triangles are similar, then by definition of similarity, that
the corresponding angles are congruent and the corresponding sides are
proportional, i. e. ABCDEF, if and only A  D, B  E, C  F,
and
AB BC CA
DE
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DF
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FD
More examples on the board
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11.2c Similar Triangles
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If ABC is a right triangle, and CD is the altitude, then the segment CD
divides the ABC into 3 similar right triangle, i.e. ABC  ACD 
BCD, and CD2 = AD*BD
C
Example:
A
D
B
If ABC is a right triangle with C = 90°, B = 50°, what about ACD,
BCD and B? If BD = 9, AD = 16, CD = ? BC = ? AC = ?
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Homework #70, pp. 591-592, #1-15
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11.3 Indirect Measurement with Similar Triangles
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Objective: Learn how to use similar triangles to measure tall objects
and large distances indirectly.
Indirect measurement means to calculate the height of tall objects
that you can’t reach using similar triangles.
Example A: A person is 5’3” tall cast a shadow of 6’. A tree is tall
and cast a shadow 18’. Find the height of the tree.
Example B: A brick building casts a shadow 7’. At the same time, a
3’ child casts a shadow of 6” long. How tall is the building?
C explain this on the board
Example C:
D
A
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E
B
Homework # 71, pp 601-602, #1-7, 11-13
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11.4 Corresponding Parts of Similar Triangles
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Objective: To investigate the relationship between two similar
triangles
Proportional Parts Conjecture: If two triangles are similar, then
the lengths of the corresponding altitudes, medians, and angle
bisectors are proportional.
Reminder: An angle bisector is not necessary a median, and a
median is not necessary an angle bisector.
Angle Bisector/Opposite Side Conjecture: A bisector of an
angle in a triangle divides the opposite side into two segments
whose lengths are in the same ratio as the lengths of the two sides
forming the angle.
Example: Prove that the lengths of the corresponding medians of
similar triangles are proportional to the lengths of the
corresponding sides.
HW#72, pp 605-607, 1-15 odd, 17
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11.5 Proportions with Area
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Objective: to discover the relationship between the area of
similar figures.
Proportional Areas Conjecture: If corresponding side lengths
of two similar polygons or the radii of two circles compare in the
ratio m/n, then their areas compare in the ratio (m/n)2 or m2/n2
Example 1: If the area of the triangle ABC has proportion to the
trapezoid BCDE is 1 to 24, find the ratio of AB/BD.
Can this conjecture apply to the surface areas? Read the
Investigation 2 on the Text.
Example 2: If you need 3 oz of shredded cheese to cover a
medium 12 in. diameter pizza, how much shredded cheese
would you need to cover a 16 in. diameter pizza?
HW#73, pp 610-612, 1-8, 10-16 even.
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11.6 Proportions with Volume
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Objective: to discover the relationship between the volumes of similar figures.
Example A: Are these right rectangular prisms similar? The smaller one has
dimension 2, 3, and 7. The larger one has dimension 2, 6, 14.
Example B: Are the right circular cones similar? The smaller one with radius
8cm, height 14, the larger one with radius 12, height 21.
Example C: Are the two right cylinders similar? The smaller one with radius
3cm, height 7cm,and the larger one with base circumference 18cm, height
21cm.
Proportional Volume Conjecture: If corresponding edge lengths (or radii, or
heights) of two similar solids compare in the ratio m/n, then their areas compare
in the ratio (m/n)3 or m3/n3
If the pentagonal pyramids are similar with h/H = 4/7, and the smaller pyramid
has volume 320cm3, what is the volume of large pyramid?
A “square can” is a right cylinder that has height equal to its diameter. One
square can has height 5cm, and another has height 12 cm. About how many full
cans of water from the smaller can are needed to fill the larger one.
HW#74, pp 616-617, 1-11 odd.
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11.7 Proportional Segments between Parallel Lines
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Objective:
 Discover the relationship in the lengths of segments formed when one or
more lines parallel to one side of a triangle intersect the other two sides.
 Learn the applications of the relationships.
Example A: Given ABC with the segment DE parallel to the side of AB, then
ABC  DEC (on the board)
Example B: From example A, if CD = 48, DA = 36, CE = 60, then EB = ?
Parallel/Proportionality Conjecture: If a line parallel to one side of a triangle
pass through the other two sides, then it divides the other two sides
proportionally. Conversely, if a line cuts two sides of a triangle proportionally,
then it is parallel to the third side.
Examples on the board.
Extended parallel/Proportionality Conjecture: IF two or more lines pass
through two sides of a triangle parallel to the third, then they divide the two
sides proportionally.
Examples on the board
HW#75, pp 627-629, 1-21 odd.
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