Download Chapter 7 Power Point Slides File

Chapter 7
Section 7.1 – Ratios and Proportions
Objectives: To write ratios and solve proportions
 Ratio -> comparison of two quantities
𝑎
 This can be written as a : b or when b ≠ 0 (for our purposes we
𝑏
will assume all terms and expressions are nonzero)
 Proportion -> a statement that two ratios are equal
 Can be written is these forms:
𝑎
𝑏
𝑐
𝑑
= and a : b = c : d
 Extended Proportion -> when three or more ratios are equal
 Properties of Proportions

𝑎
𝑏
𝑐
=
𝑑
is equivalent to:
1. ad = bc
2.
3.
4.
𝑏 𝑑
=
𝑎 𝑐
𝑎 𝑏
=
𝑐 𝑑
𝑎+𝑏 𝑐+𝑑
=
𝑏
𝑑
means
a:b=c:d
extremes
Cross Product Property -> also know as “the
product of the extremes is equal to the
product of the means”
 Scale drawing -> a given scale compares each length in the
drawing to the actual length. The lengths used may be
different units, possibly written as 1 in. to 100 mi, 1 in. = 12
ft, or 1 mm : 1 m.
 Examples:
 Solve the following proportions
𝑥
5
12
=
7
5 20
=
𝑧
3
18
6
=
𝑛+6 𝑛
 Write two proportions that are equivalent to:
𝑥
𝑦
=
5
6
𝑚
4
=
𝑛
11
Homework # 1
Due Wednesday (January 09)
Page 368 – 369
# 1 – 19 odd
# 35 – 42 all
Section 7.2 – Similar Polygons
 Objectives: To identify similar polygons
To apply similar polygons
 Similar (~) -> two figures that have the same shape but not
necessarily the same size
 1. Corresponding angles must be congruent
 2. Corresponding sides are proportional
 Similarity Ratio -> the ratio of the lengths of corresponding
sides
 ABCD ~ EFGH. Complete each statement.
B
C
F
G
127°
E
53°
A
D
H
𝐴𝐵
𝐸𝐹
m<E=?
m<B=?
=
𝐴𝐷
?
 Determine whether the triangles are similar. If they are,
write a similarity statement and give the similarity ratio.
B
15
A
12
18
E
C
20
16
D
24
F
 LMNO ~ QRST
O
2
Find the value of x.
N
T
3.2
L
5
x
S
M
Q
6
R
 Golden Rectangle -> a rectangle that can be divided into a
square and a rectangle that is similar to the original
rectangle.
 Golden Ratio -> the ratio in any golden rectangle
about 1.618 : 1
Homework #2
Due Friday (January 11)
Page 375 – 376
# 1 – 16 all
# 21 – 29 odd
Section 7.3 – Proving Triangles Similar
 Objectives: To use AA, SAS, and SSS similarity statements
To apply AA, SAS, and SSS similarity
statements
 Postulate 7.1
 Angle-Angle Similarity (AA ~) Postulate
 If two angles of one triangle are congruent to two
angles of another triangle, then the triangles are similar.
P
T
R
S
L
M
 Theorem 7.1 -> Side-Angle-Side Similarity (SAS ~)
Theorem
 If an angle of one triangle is congruent to an angle of a
second triangle, and the sides including the two angles are
proportional, then the triangles are similar.
 Theorem 7.2 -> Side-Side-Side Similarity (SSS ~)
Theorem
 If all three corresponding sides of two triangles are
proportional, then the triangles are similar.
 Ex: Explain why the triangles must be similar.
Write a similarity statement.
Z
P
6
R
8
3
Q
Y
4
X
 Ex: Explain why the triangles must be similar.
Write a similarity statement.
9
A
6
B
G
6
8
8
C
E
12
F
Homework # 3
Due Monday (January 14)
Page 385 – 387
# 1 – 19 odd
# 24 – 27 all
Section 7.4 – Similarity in Right
Triangles
 Objectives: To find and use relationships in similar
right triangles
 Theorem 7.3
 The altitude to the hypotenuse of a right triangle divides the
triangle into two triangles that are similar to the original triangle
and to each other.
C
A
Triangle ABC ~ Triangle ACD ~ Triangle CBD
B
D
 Geometric Mean -> proportions in which the means are equal.
For any two positive numbers a and b, the geometric mean of
𝑎 𝑥
a and b is the positive number x such that =
𝑥
𝑏
**We will cover two corollaries that involve a geometric mean**
 Corollary 1 to Theorem 7.3
 The length of the altitude to the hypotenuse of a right triangle is
the geometric mean of the lengths of the segments of the
hypotenuse.
C
𝐴𝐷
𝐶𝐷
A
D
B
𝐶𝐷
=
𝐷𝐵
 Corollary 2 to Theorem 7.3
 The altitude to the hypotenuse of a right triangle separates the
hypotenuse so that the length of each leg of the triangle is the
geometric mean of the length of the adjacent hypotenuse
segment and the length of the hypotenuse.
C
A
D
B
𝐴𝐷
𝐴𝐶
𝐴𝐶
=
𝐴𝐵
𝐷𝐵
𝐶𝐵
𝐶𝐵
=
𝐴𝐵
Homework #4
Due Tuesday (January 15)
Page 394
# 1 – 20 all
# 26 – 33 all
Section 7.5 – Proportions in Triangles
 Objectives:
To use the Side-Splitter Theorem
To use the Triangle-Angle-Bisector Theorem
 Theorem 7.4 -> Side-Splitter Theorem
 If a line is parallel to one side of a triangle and intersects the
other two sides, then it divides those sides proportionally.
Q
R
X
S
𝑋𝑅
𝑅𝑄
Y
𝑌𝑆
=
𝑆𝑄
 Corollary to Theorem 7.4
 If three parallel lines intersect two transversals, then the
segments intercepted on the transversals are proportional.
𝑎
𝑏
a
b
c
d
𝑐
=
𝑑
 Theorem 7.5 -> Triangle-Angle-Bisector Theorem
 If a ray bisects an angle of a triangle, then it divides the opposite
side into two segments that are proportional to the other two
sides of the triangle.
𝐶𝐷
𝐷𝐵
A
B
C
D
𝐶𝐴
=
𝐵𝐴
Homework # 5
Due Wednesday (Jan 16)
Page 400 – 401
# 1 – 24 all
Quiz Friday
Test Thursday/Friday next week