Transcript 8.2 Problem Solving in Geometry with Proportions Geometry Mrs. Spitz Spring 2005 Objectives/Assignment   Use properties of proportions Use proportions to solve real-life problems such as using the scale of a map. Slide.

8.2 Problem Solving in
Geometry with
Proportions
Geometry
Mrs. Spitz
Spring 2005
Objectives/Assignment


Use properties of
proportions
Use proportions to
solve real-life problems
such as using the scale
of a map.
Slide #2




Pp. 2-32 all
Reminder: Quiz after
8.3.
Ch. 8 Definitions due
Ch. 8
Postulates/Theorems
due
Using the properties of
proportions


In Lesson 8.1, you studied the reciprocal
property and the cross product property. Two
more properties of proportions, which are
especially useful in geometry, are given on
the next slides.
You can use the cross product property and
the reciprocal property to help prove these
properties.
Slide #3
Additional Properties of
Proportions
IF
a
b
=
c ,
d
then
a
c
=
b
d
IF
a
b
=
c ,
d
then
Slide #4
a+b
b
=
c+d
d
Ex. 1: Using Properties of
Proportions
IF
p
6
p
6
p
r
=
=
=
r ,
10
r
p
r
then
=
3
5
Given
10
6
10
a
b
Slide #5
=
c
a
b
=
,
then
d
c
d
Ex. 1: Using Properties of
Proportions
IF
p
r
=
3
5
Simplify
 The statement is true.
Slide #6
Ex. 1: Using Properties of
Proportions
a
3
=
a+3
3
a+3
3
c
4
c+4
=
≠
4
c+4
4
Given
a
c
a+b c+d
=
=
,
then
b
d
b
d
Because these
conclusions are not
equivalent, the
statement is false.
Slide #7
Ex. 2: Using Properties of
Proportions

In the diagram
A
AB
AC
=
BD
CE
16
B
Find the length of
30
C
x
D
BD.
Do you get the fact that AB ≈ AC?
Slide #8
10
E
A
16
30
B
C
x
10
D
E
Solution
AB = AC
BD CE
16 = 30 – 10
x
10
16 = 20
x
10
20x = 160
x=8
Given
Substitute
Simplify
Cross Product Property
Divide each side by 20.
Slide #9
So, the length of BD is 8.
Geometric Mean

The geometric mean of two positive numbers
a and b is the positive number x such that
a
x
=
x
b
If you solve this
proportion for x, you
find that x = √a ∙ b
which is a positive
number.
Slide #10
Geometric Mean Example

For example, the geometric mean of 8 and 18
is 12, because
8
12
=
12
18
and also because x = √8 ∙ 18 = x = √144 = 12
Slide #11
Ex. 3: Using a geometric mean

PAPER SIZES.
International standard paper
sizes are commonly used
all over the world. The
various sizes all have the
same width-to-length ratios.
Two sizes of paper are
shown, called A4 and A3.
The distance labeled x is
the geometric mean of 210
mm and 420 mm. Find the
value of x.
Slide #12
A4
x
A3
210 mm
x
420 mm
A4
x
A3
420 mm
210 mm
x
Solution:
210
x
=
The geometric mean of 210 and 420 is 210√2, or about 297mm.
x
Write proportion
420
X2 = 210 ∙ 420
Cross product property
X = √210 ∙ 420
Simplify
X = √210 ∙ 210 ∙ 2
X = 210√2
Slide #13
Factor
Simplify
Using proportions in real life

In general when solving word problems that
involve proportions, there is more than one
correct way to set up the proportion.
Slide #14
Ex. 4: Solving a proportion

MODEL BUILDING. A scale model of the
Titanic is 107.5 inches long and 11.25 inches
wide. The Titanic itself was 882.75 feet long.
How wide was it?
Width of Titanic
Width of model
Length of Titanic
=
Length of model
LABELS:
Width of Titanic = x
Width of model ship = 11.25 in
Length of Titanic = 882.75 feet
Length of model ship = 107.5 in.
Slide #15
Reasoning:
Width of Titanic
Width of model
=
Length of Titanic
Length of model
x feet
11.25 in.
Write the proportion.
882.75 feet
=
x=
107.5 in.
11.25(882.75)
107.5 in.
Substitute.
Multiply each side by 11.25.
Use a calculator.
x ≈ 92.4 feet
Slide #16
So, the Titanic was about 92.4
feet wide.
Note:


Notice that the proportion in Example 4
contains measurements that are not in the
same units. When writing a proportion in
unlike units, the numerators should have the
same units and the denominators should
have the same units.
The inches (units) cross out when you cross
multiply.
Slide #17