GEOMETRY
MODULE 2 LESSON 2
MAKING SCALE DRAWINGS USING THE RATIO METHOD
OPENING EXERCISE
Please read the following information about dilation.

A dilation is a rule (a function) that moves points in the plane a specific distance along the ray
that originates from a center O. The location of the scaled point is determined by the scale
factor and the distance of the original point form the center.

The scale factor for a dilation where a point is pulled toward the center must be 0 < 𝑟 < 1.

The scale factor for a dilation where a point is pushed away from the center must be 𝑟 > 1.

A point, different from the center, that is unchanged in its location after a dilation must have a
scale factor of 𝑟 = 1.

Scale factors are always positive values, as we use it when working with distance. (If we were
to use negative values for scale factors, we would be considering distance as a negative value,
which does not make sense.)
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PRACTICE
1. Create a scale drawing of the figure below using the ration method about center O and scale
1
factor 𝑟 = 2.
Step 1: Draw a ray beginning at O through each vertex of the figure.
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Step 2: Dilate each vertex along the appropriate ray by scale factor 𝑟 = 2. Use a ruler to find
the midpoint between O and D and then each of the other vertices. Label each respective
midpoint with prime notation.
Step 3: Join the vertices in the way they are joined in the original figure.

Why did we locate the midpoint of O and D?
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The scale factor is 2. Midpoint is located at the half-way point.

Use patty paper to compare the angles in ∆𝐸𝐷𝐶 to the angles in ∆𝐸′𝐷′𝐶′. Are they the same
measure?
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2. Create a scale drawing of the figure below using the ration method about center O and scale
factor 𝑟 = 3.
Step 1: Draw a ray beginning at O through each vertex of the figure.
⃗⃗⃗⃗⃗ ; 𝐴′ should be three times as far
Step 2: Use your ruler to determine the location of 𝐴′ on 𝑂𝐴
from O as A. Determine the locations of 𝐵’ and 𝐶’ in the same way along the respective rays.
Step 3: Draw the corresponding line segments.
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ON YOUR OWN
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Use the figure below with center O and a scale factor of 𝑟 = 2 to create a scale drawing. Compare
your work with a classmate and verify that the corresponding angles are equal in measure.
DISCUSSION
A clothing company wants to print the face of the Statue of Liberty on a T-shirt. The length of the
face from the top of the forehead to the chin is 17 feet, and the width of the face is 10 feet. Given
that a medium-sized T-shirt has a length of 29 inches and a width of 20 inches, what dimensions of
the face are needed to produce a scaled version that will fit on the T-shirt?
a. What shape would you use to model the face of the statue?
Any shape with strong vertices would work best: Triangle, Rectangle. Oval and Circle could
work as well.
b. Knowing that the maximum width of the T-shirt is 20 inches, what scale factor is needed to
make the width of the face fit on the shirt?
𝑠𝑐𝑎𝑙𝑒𝑑
20
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The units must be converted from feet to inches. 𝑟 = 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 = 120 = 6
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c. What scale factor should be used to scale the length of the face?
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A dimensions of well-scaled image must be proportional, so 6 must be the scale factor.
d. Using the scale factor identified in part ©, what is the scaled length of the face? Will it fit on
the shirt?
1
= 34 𝑖𝑛𝑐ℎ𝑒𝑠.
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Using this scale factor, the image will not fit on the t-shirt.
204 𝑖𝑛𝑐ℎ𝑒𝑠 ×
e. Identify the scale factor you would use to ensure that the face of the statue was in proportion
and would fit on the T-shirt. Identify the dimensions of the face that will be printed on the shirt.
Length: 204 ×
Length: 204 ×
1
7
= 29.1 still too big
1
= 25.5 will fit Width 120 ×
8
1
8
= 15
f. The T-shirt company wants the width of the face to be no smaller than 10 inches. What scale
factor could be used to create a scaled version of the face that meets this requirement?
Width: 120 ×
1
= 10 So any scale factor between
12
1
8
1
and 12.
g. If it costs the company $0.005 for each square inch of print on a shirt, what is the maximum and
minimum costs for printing the face of the Statue of Liberty on one T-shirt?
Largest Area: 15 × 25.5 = 382.5 𝑠𝑞𝑢𝑎𝑒 𝑖𝑛𝑐ℎ𝑒𝑠
Cost: 382.5 × 0.005 = $1.91
Smalles Area: 10 × 17 = 170 𝑠𝑞𝑢𝑎𝑒 𝑖𝑛𝑐ℎ𝑒𝑠
Cost: 170 × 0.005 = $0.85
SUMMARY

To create a scale drawing using the ration method, each vertex of the original figure is dilated
about the center O by scale factor r. Once all the vertices are dilated, they are joined to each
other in the same way as in the original figure.

The scale factor tells us whether the scale drawing is being enlarged 𝑟 > 1 or
reduced 0 < 𝑟 < 1.
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