FUNDAMENTALS
Name _________________
A.34 Ratios and Proportions using Shapes
We saw yesterday that ratios and proportions can help us to design buildings, properly administer
medicine, etc. Suppose we want to use this idea of similar shapes to make larger designs from smaller
ones (like parents decorating a nursery who project small drawings onto a wall before painting the
larger design), or model train enthusiasts who use the actual scale of a locomotive to manufacture
small-scale model trains.
E
Let’s use an easy shape like a triangle:
B
A
B
A
C
D
F
Triangle ABC is similar to triangle DEF (that is, they have the same shape and their lengths are based
on ratios). Suppose the length of side AB is 3 (mark this on your paper). The length of side AC is 4.
The length of side BC is 5.
We are told that the ratio of the 2nd triangle to the 1st one is 2:1 (which means that each side of
triangle DEF is twice as large as triangle ABC.
DEF 2
 2
ABC 1
DE is similar to which side? ______ What will be the length of side DE? DE = 2  ____ = _______
EF is similar to which side? ______ What will be the length of side EF? EF = 2  ____ = _______
DF is similar to which side? ______ What will be the length of side DF? DF = 2  ____ = _______
B
A
B
Now let’s look at some pentagons (5 sides)
G
B
C
A
E
F
H
D
J
I
The ratio of the smaller pentagon to the larger one is 1:3
If the length of each side of the small pentagon is 6 cm, what is the length of each side of the large
pentagon?
Set up a ratio to solve:
Homework: A.34
1. For a square, the length of all sides are ___ ____ ____ ____ ____ (fill in the blank with a word).
The ratio of the length of each side of a small square to the length of the side of a larger square is
3
(or 3 : 4)
4
If the small square has a side of length 12 inches, what is the length of the side for the larger square?
2. Rail transport modeling uses a variety of scales (ratio between the real world and the model) to
ensure scale models look correct when placed next to each other. Model railway scales are standardized
worldwide by many organizations and hobbyist groups. Scales may be expressed as a numeric ratio (e.g.
1/87 or 1:87) or as a letter defined in rail transport modelling standards (e.g. HO, OO, N, O, G, TT and Z.)
(courtesy of Wikipedia)
N scale is a popular model railway scale. Depending upon the manufacturer (or country), the scale
ranges from 1:148 to 1:160. Using a ratio of 1:160, ( that is, N/Real = 1/160) if a real locomotive has a
length of 320 feet and a height of 20 feet, what would be the dimensions of the N scale model?
N  scale
1

Real
160
Height of N-scale model = _____
Length of N-scale model = _____
More practice with proportions and ratios:
Solve each proportion by using cross products.
3.
x 5

31 3
4.
7.
5
8

x5 3
8.
11 22

x
6
5( x  1) 2 x  11

40
4
5.
x 8

9 24
6.
9.
2
8

10 2x
3 x  1 6 x  30

4
12