MOD2 L5 1

GEOMETRY

MODULE 2 LESSON 5

SCALE FACTORS

OPENING EXERCISE

In each of the figures below, ∆𝐴𝐵𝐶 has been dilated from the center O by some scale factor to

produce the image ∆𝐴′𝐵′𝐶′. Describe how each of the figures have been transformed and state a

scale factor.

Figure 1 has a scale factor of 1. Figure 2 has a scale factor greater than 1. Figure 3 has a scale factor less than 1.

MOD2 L5 2

DISCUSSION

Dilation Theorem: If a dilation with center O and scale factor r sends point P to P’ and Q to Q’,

then 𝑃′𝑄′ = 𝑟 𝑃𝑄 .

Furthermore, if 𝑟 ≠ 1 and O, P, and Q are vertices of a triangle, then 𝑃𝑄 ∥ 𝑃′𝑄′.

The dilation theorem state two things:

1. If two points, P and Q, are dilated from the same center using the same scale factor, then the

segment formed when you connect the dilated points P’ and Q’ is exactly the length of 𝑃𝑄

multiplied by the scale factor.

2. The lines containing the segments P’Q’ and PQ are parallel or equal.

For example, if points P and Q are dilated from center O by a scale factor of 𝑟 = !!, then the lines

containing the segments P’Q” and PQ are parallel, and 𝑃’𝑄’ = !!𝑃𝑄, as shown below.

𝑃’𝑄’ =32𝑃𝑄

𝑃’𝑄’ =32 5 =

152 = 7.5

MOD2 L5 3

PRACTICE

1. Produce a scale drawing of ∆𝐷𝐺𝐻 using either the ratio or parallel method with point M as the

center and a scale factor of 2 .

2. Given the diagram below, determine if ∆𝐷𝐸𝐹 is a scale drawing of ∆𝐷𝐺𝐻.

Explain why or why not? Is 𝐸𝐹 ∥ 𝐺𝐻?

𝐷𝐸 = 3.2𝑚𝑚, 𝐸𝐺 = 3.75𝑚𝑚, 𝐸𝐹 = 5.9𝑚𝑚, 𝐺𝐻 = 11.9𝑚𝑚

𝐷𝐸 = 𝑟𝐷𝐺

3.2 = 𝑟(3.2+ 3.75)

𝑟 =3.26.95 = 0.46

𝐸𝐹 = 𝑟𝐺𝐻

5.9 = 𝑟(11.9)

𝑟 =5.911.9 = 0.496

The scale factors are not the same. Therefore ∆𝐷𝐸𝐹 is not a scale drawing of ∆𝐷𝐺𝐻 and 𝐸𝐹 ∦ 𝐺𝐻.

MOD2 L5 4

3. ∆𝐴𝐵′𝐶′ is a dilation of ∆𝐴𝐵𝐶 from vertex A, and 𝐶𝐶’ = 2. Use the given information and the

diagram to find 𝐵′𝐶′.

• 𝐴𝐶 = 4 and 𝐵𝐶 = 7

• ON YOUR OWN: 𝐴𝐶 = 7 and 𝐵𝐶 = 9

𝐴𝐶′ = 𝑟𝐴𝐶

(4+ 2) = 𝑟(4)

𝑟 =64 =

32 = 1.5

𝐵𝐶′ = 𝑟𝐵𝐶

𝐵𝐶! =32 7 𝑜𝑟 (1.5)(7)

𝐵𝐶! =212 𝑜𝑟 10.5

𝐴𝐶′ = 𝑟𝐴𝐶

(7+ 2) = 𝑟(7)

𝑟 =97 = 1.286

𝐵𝐶′ = 𝑟𝐵𝐶

𝐵𝐶! =97 9 𝑜𝑟 (1.286)(9)

𝐵𝐶! = 817 𝑜𝑟 11.574

SUMMARY

• Dilation Theorem: If a dilation with center O and scale factor r sends point P to P’ and Q to Q’,

then 𝑃′𝑄′ = 𝑟 𝑃𝑄 . Furthermore, if 𝑟 ≠ 1 and O, P, and Q are vertices of a triangle, then

𝑃𝑄 ∥ 𝑃′𝑄′.

• Three methods for scale drawings:

o Compass Construction

o Ratio Method (Using Dilation)

o Parallel Method