Download Algebra 3.6 Notes

3.6 Solve Proportions Using Cross Products
Warm-up: Follow the directions below the table.
STEP 1 Determine whether the pairs of ratios in the first column are equivalent. Write yes or no in
the second column and explain how you know in the third column.
STEP 2 For each pair of ratios in the table find the product on each diagonal and record the products
in the table as follows: Multiply the number in position a by the number in position d and record in
column A. Multiply the number in position b by the number in position c and record in column B.
a
c
and
b
d
Then compare column A with column B and decide if they are equal.
RATIO
EQUIVALENT?
HOW DO YOU KNOW?
A
B
1
5
and
2
10
2
6
and
7
21
4
8
and
9
18
1
11
and
3
30
3
4
and
15
20
Yes
1 x 5 = 5 and 2 x 5 = 10
10
10
A and B
Equal?
YES
STEP 3 Look back at your table and answer the following:
When the ratios are equivalent, what is true about the product of the numbers on each diagonal?
When the ratios are not equivalent, what is true about the product of the numbers on each diagonal?
If
a
b
= cd
where b is not equal to 0 and d is not equal to 0, then what must be true about
the products ad and bc?
1
{2, 4, 5, 10}
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Step 4: Use the following four numbers to write three different proportions:
Goal  Solve proportions using cross products.
VOCABULARY:
scale drawing:
scale model:
scale:
similar triangles:
CROSS PRODUCTS PROPERTY
Words
The cross products of a proportion are ___________________
Example 5 10

6 12
_____  10 = 60
_____  12 = 60
Algebra If a  c where b ≠ 0 and ≠ 0, then ad = _____
b d
Example 1 - Solve a proportion using cross products
8 6

x 15
CHECK!
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2
Example 2 - Solve a proportion using the distributive property
4
8

CHECK!
x x3
Example 3 - Application
SEALS Each day, the seals at an aquarium are each fed 8 pounds of food for every 100 pounds of
their body weight. A seal at the aquarium weighs 280 pounds. How much food should the seal be fed
per day ?
Step 1 Write a proportion involving two ratios that compare the amount of the food with the weight
of the seal:
8
x

amount of plant food
100 280
weight of seal
Step 2 Solve the proportion.
8
x

100 280
A 280 pound seal should be fed ________ pounds of food a day.
Now You Try It! Solve the proportion. Check your solution.
1) 4  24
a 30
2)
3 = 2
x x  6
3) In Example 3, suppose the seal weighs 260 pounds. Write and solve
a proportion to find out how much food should the seal be fed each day.
Example 4 - Use a scale model
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The distance is:
3
Use a metric ruler and the map of Ohio to estimate the distance between Cleveland and Cinncinati.
From the map’s scale you can see that 1 cm = 85 km. Assume the distance with a ruler is 4.2 cm.
Example 5 – Similar Triangles
Similar triangles have proportional sides. Find the missing side lengths of these similar triangles:
B
8 cm
9 cm
E
C
12 cm
13.5 cm
A
12 cm
F
?
D
ABC ~
DEF
Use a proportion to find the length of side DF.
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4
Example 6 – Indirect measurement
Use indirect measurement to find the height of an object. The girl is standing 8 feet from the
mirror; the tree is 18 feet from the mirror. If the distance from the floor to the girls eyes is 4 feet 8
inches, how tall is the tree?