Download UNIT 7 - Peru Central School

Geometry UNIT 7
Proportions and Similarity
Day 1
Ratio/Proportion and Means/Extremes
Day 2
Similar Polygons
Day 3
Similar Triangles
Day 4
Parallel Lines and Proportional Parts
Day 5
QUIZ REVIEW
Day 6
QUIZ
Day 7
Parts of Similar Triangles
Day 8
Similarity Transformations
Day 9
Scale Drawings and Models
Review
TEST
Vocabulary
Ratio
Means
Similar Polygons
Dilation
Enlargement
Scale Drawing
Extended Ratio
Extremes
Scale Factor
Similarity Transformation
Reduction
Page 1
Proportion
Cross Products
Midsegment
Center of Dilation
Scale Model
Ratio & Proportions
WRITE & USE RATIOS
A ratio is a comparison of two quantities using division. The ratio of quantities a and b can be
expressed as a to b, a:b, or
Example:
a
, where b  0. Ratios are usually expressed in simplest form.
b
SPORTS – A baseball player’s batting average is the ratio of the number of base hits
to the number of at-bats, not including walks. According to ESPN MLB, Detroit
Tigers Miguel Cabrera had the highest batting average in Major League Baseball in
2013. If he had 555 official at-bats and 193 hits, find his batting average. Round the
answer to the nearest thousandth.
Example: In Peru High School, if there are 73 teachers and 758 students. What is the
approximate student-teacher ratio at our school?
Extended ratios can be used to compare three or more quantities. The expression a:b:c means
that the ratio of the first two quantities is a:b, the ratio of the last two quantities is b:c and the ratio
of the first and the last quantities is a:c.
Example:
The ratio of the measures of the angles in  ABC is 3:4:5. Find the measures of the
angles.
3x°
5x°
Page 2
4xo
USE PROPERTIES OF PROPORTIONS
An equation stating two ratios are equal is called a proportion. In the proportion
a c
 , the
b d
numbers a and d are called the extremes of the proportion, while the numbers b and c are called
the means of the proportion.
a c

b d
The product of the extremes ad and the means bc are called cross products.
THE PRODUCT OF THE MEANS = THE PRODUCT OF THE EXTREMES
Example: CAR OWNERSHIP – Fernando conducted a survey of 50 students and found that
28 owned their own cars. If 755 students drive to his school, predict the number of students with
their own cars.
PERU
Example: Solve for x:
Example:
Solve for x:
3x – 1 =
4
2x + 4
5
x 2 – 8x + 8 =
20
Page 3
-x + 2
5
PRACTICE:
1.
2.
3.
4.
5.
Find the value of x.
Page 4
6.
Find the value of x.
7.
8.
Find x and round to the nearest tenth if necessary.
9.
Page 5
Similar Polygons
People often customize their computer desktops using photos, centering the images at their original
size or stretching them to fit the screen. This second method distorts the image, because the
original and new images are not geometrically similar.
Similar polygons have the same shape, but not necessarily the same size. (Also known as
the similarity ratio.)
 If two polygons are similar, then their perimeters are proportional to the scale factor
between them.
The ratio of the lengths of the corresponding sides of two similar polygons is called the
scale factor. The scale factor depends on the order of comparison. If a specific order is not
given, scale factor is the ratio of the image to the pre-image.
In the diagram,  ABC ~  XYZ.
A
C
Y
The scale factor of  ABC to  XYZ is
6
or 2.
3
The scale factor of  XYZ to  ABC is
3
1
or .
6
2
a.
b.
Page 6
6
3
B
X
Z
Example:
Find the value of each variable if  JLM ~  QST.
J
4
S
M
3
3y – 2
5
6x – 3
T
2
Q
L
Page 7
PRACTICE:
1.
2.
3.
4.
Find the perimeter of:
Page 8
5.
**Determine if all angles are 90o by using slope formula and showing perpendicular lines.
**Find the length of each side (distance formula) and determine if corresponding sides are
proportional.
Page 9
Similar Triangles
 Angle-Angle (AA) Similarity – If two angles of one triangle are similar to two angles of
another triangle, then the triangles are similar.
Example: Determine whether the triangles are similar. If so, write a similarity statement.
Explain your reasoning.
a.
A
C
B
b.
J
47°
P
44°
L
Example:
Q
K
Find each measure.
a) QP and MP
b) WR and RT
M
S
5
Q
x
P
6
8
N
W
3
3
5
x+6
2x + 6
R
10
V
O
Page 10
T
Page 11
Page 12
Parallel Lines & Proportional Parts
 Triangle Proportionality Theorem – If a line is parallel to one side of a triangle and intersects
the other two sides, then it divides the sides into segments of proportional lengths.
Example:
In  PQR, ST || RQ . If PS = 12.5, SR = 5 and PT = 15, find TQ.
P
T
Q
S
R
 Converse of Triangle Proportionality Theorem – If a line intersects two sides of a triangle
and separates the sides into proportional corresponding segments, then the line is parallel to the
third side of the triangle.
E
Example: DG is half the length of GF, EH = 6, and
HF = 10. Is DE || GH ?
D
H
G
F
A midsegment of a triangle is a segment with endpoints that are the midpoints of two sides of
the triangle. Every triangle has three midsegments.
 Triangle Midsegment Theorem – A midsegment of a triangle is parallel to one side of the
triangle, and its length is one half the length of that side.
Page 13
Example:
Find each measure.
a) DE = _____
b) DB = _____
c) m  FED = _____
A
15
F
D
9.2
C
82°
E
B
 Proportional Parts of Parallel Lines – If three or more parallel lines intersect two transversals,
then they cut off the transversals proportionately.
Example:
REAL ESTATE – Frontage is the measurement of a property’s boundary that runs
along the side of a particular feature such as a street, lake, ocean, or river. Find the
ocean frontage for Lot A to the nearest tenth of a yard.
Ocean
Lot A
60 yd.
Lot B
 Congruent Parts of Parallel Lines – If three or more parallel lines cut off congruent segments
on one transversal, then they cut off congruent segments on every transversal.
Example:
Page 14
PRACTICE:
1.
2.
3.
Page 15
4.
5.
6.
Page 16
QUIZ REVIEW
SHOW WORK for #1 – 5 below (if necessary):
Page 17
SHOW WORK for #1-5 below (if necessary):
Page 18
Parts of Similar Triangles
 Special Segments in Similar Triangles
Label the following special segments as median, altitude, or angle bisector. Write a mathematical
sentence below the label to prove you are correct!
M
A
T
S
H
S
A
W
E
O
M
E
If two triangles are similar the lengths of the corresponding altitudes are proportional to the lengths
of the corresponding sides. The same applies to angles bisectors and medians.
Special note: In an equilateral triangle, the altitude, the angle bisector and the median are all the
same line. In a scalene triangle, the altitude, the angle bisector and the median are all different lines.
Try drawing all three lines on the triangles below.
Example:
In the figure,  KLM ~  XZY. Find the value of x.
K
P
L
Y
20
16
M
Page 19
x
Z
Q
15
X
 Triangle Angle Bisector – An angle bisector in a triangle separates the opposite side into two
segments that are proportional to the lengths of the other two sides.
Example:
Find x.
x
13
4
Example:
6
Find x.
14
11
x
20
Page 20
Page 21
PRACTICE:
1.
2.
3.
Find x.
Page 22
Similarity Transformations
A transformation is an operation that maps an original figure, the preimage, onto a new figure
called the image.
A dilation is a transformation that enlarges or reduces the original figure proportionality. Since a
dilation produces a similar figure, a dilation is a type of similarity transformation.
Dilations are performed with respect to a fixed point called the center of dilation.
The scale factor of a dilation describes the extent of the dilation. The scale factor is the ratio of a
length on the image to a corresponding length on the preimage.
The letter k usually represents the scale factor of a dilation. The value of k determines whether the
dilation is an enlargement or a reduction.
3.
Page 23
Page 24
PRACTICE:
1.
2.
Page 25
Scale Drawings and Models
A scale model or a scale drawing is an object or drawing with lengths proportional to the object
it represents. The scale of a model or drawing is the ratio of a length on the model or drawing to
the actual length of the object being modeled or drawn.
Examples:
Page 26
4.
Page 27
Page 28
Page 29