Transcript Chapter 8 Notes - Troy High School

8.1 – Ratio and Proportion
Solve Proportions
Reduce Ratios
Find unknown lengths given ratios
If a and b are two quantities that are measured in the same units,
then the ratio of a to b is a . It can also be written as a:b.
b
Because a ratio is a quotient, the denominator cannot be zero.
You also reduce ratios when possible, like 2:4 being 1:2
Given that there are 18 boys and 15 girls in class.
Write the ratio of guys to girls
Write the ratio of girls to guys
Write the ratio of girls to students
Does order matter? Why or why not?
To help with units:
1 ft = 12 in,
3 ft = 1 yard
5280 ft = 1 mi,
100 cm = 1 m
1000 g = 1 kg,
16 oz = 1 lb
1000 m = 1 km
Generally, I’d
convert to smaller
units
Simplify
30 ft
2 yd
Simplify
3 lb
26 oz
Ratio between the length and width of a rectangle is 3:2. The
perimeter is 30. Find the length of each side
What do they want you to find?
What information is given?
What formula do you think you’ll need?
How can you represent the information you need to find?
Solve the problem
What steps do you think are required to solve these types of problems?
Ratio between to supplementary angles is 10:8
Ratio of three sides of a triangle is 5:6:7. The perimeter is 54.
Ratio of angles of a triangle is 1:2:3.
When you equate two ratios, it’s called a proportion.
a c

b d
Extremes – First and last terms
Means – Middle terms
Cross-product property
a c
  ad  bc
b d
Multiplying the m eans  m ultiplying the extrem es
ReciprocalP roperty
a c
b d
 is equivalent to 
b d
a c
Setting up Proportions.
A
D
The ratio of AB to AD is
3:5. If AB = y + 2, and
AD = 3y – 2, find y.
B
C
The ratio of AD to CD is
1:3. If AD = x + 2, and
CD = 6x, find AD.
x x 1

6
4
2x 1 4

x2 3
Challenge
y x 1

6
3
y  2 x 8

8
12
A miniature goat is 12 inches tall, while a regular goat is
42 inches tall.
If a regular goat body is
If a miniature goat’s leg
50 inches long, then
is 2 inches tall, how tall
how long is the body of
is the regular goat’s leg?
a miniature goat?
8.2 – Problem Solving in
Geometry with Proportions
Solve Proportions
Geometric Mean
Find unknown sides
a=x
x b
Find the geometric
mean between 4 and
20.
If a, b, and x are positive, x is
called the GEOMETRIC MEAN
between a and b.
Find the geometric
mean between 8 and
24.
More Properties of proportions:
a c
If

b d
a b
then 
c d
a c
If

b d
ab cd
then

b
d
2 3
2 x
If
 ; then

x y
x
2 3
?
y
If
 ; then

x y
2
3
2 3
2 ?
If
 ; then 
x y
3 y
3 5
8
If
 ; then  ?
q z
5
?
y
Decide if the statement is true or false
If
x 3
x y
 ; then 
y 4
4 3
2 3
2 x y 3
If
 ; then

x y
x
y
2 3
If
 ; then 2 x  3 y
x y
If
vw zq
v w z z  q  w

; then

z
w
z
w
N
NO NM

OL ME
O
L
E
NO OL NL NM ME NE
4
2
8
NO OL NL NM ME NE
6
9
M
12
D
AX
AZ

XD ZB
C
X
A
Y
Z
B
AX XD AD AZ ZB AB
4
3
6
AX XD AD AZ ZB AB
5
6
9
• Write down one comment or question that you
have on the material so far this chapter, you
may be randomly called on.
8.4 – Similar Triangles
8.5 – Proving Triangles are
Similar
AA Similarity Postulate – If two angles of one
triangle are congruent to two angles of
another triangle, then the triangles are similar.
B
A
C
Given : A  A' , B  B'
Then : ΔABC ~ ΔA' B' C '
SAS ~ Thrm : If an  of one
SSS ~ Thrm : If the sides
 is  to an  of another 
of 2 ' s are in proportion ,
and the sides including those
then the  ' s are ~
' s are in proportion , then
the  ' s are ~
A
AB AC
If

, A  D
B
DE DF
T henABC ~ DEF
C
AB AC BC
If


DE DF EF
T henABC ~ DEF
D
E
F
Are these triangles similar? Why? (Use
complete sentences)
12
16
9
12
22.5
12
12
30
16
16
Are these triangles similar? Why? (Use
complete sentences)
500
400
500
900
500
12
400
15
500
10
900
8
12
16
20
15
10
8
FIND THE SCALE FACTOR! Solve for the unknown variables.
D
8
10
15
8+y
U
5
C
12
z
S
y
K
8.6 – Proportions and Similar
Triangles
AL CM
P ointsL and M lie on AB and CD. If

, then
LB MD
AB and CD are DIVIDEDP ROP ORT IONALLY.
A
L
C
B
M
D
Triangle Proportionality Theorem – If a line is parallel to one
side of a triangle and intersects the other two sides, then it
divides those sides proportionally.
T
If : RST; PQ || RS
RP SQ
Then :

PT QT
P
R
Q
S
Converse of Triangle Proportionality Theorem - If the sides
are proportional, then the lines are parallel.
Theorem – If three || lines intersect two transversals, then
they divide the transversals proportionally.
R
X
If : RX || SY || TZ
RS XY
Then :

ST YZ
S
Y
T
Z
Triangle Angle-Bisector Thrm – If a ray bisects an angle of a
triangle, then it divides the opposite side into segments
proportional to the other two sides.
F
G
If : DEF; DG bis. FDE
GF DF
GF GE
Then :

or

GE DE
DF DE
D
E
x
14
3
8
x
8
y
12
20
4
12
18
x
8
12
x
9
6
15
40
x
20
y
12
20
x