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MOD2 L5 1
GEOMETRY
MODULE 2 LESSON 5
SCALE FACTORS
OPENING EXERCISE
In each of the figures below, βπ΄π΅πΆ has been dilated from the center O by some scale factor to
produce the image βπ΄β²π΅β²πΆβ². 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 Pβ and Q to Qβ,
then πβ²πβ² = π ππ .
Furthermore, if π β 1 and O, P, and Q are vertices of a triangle, then ππ β₯ πβ²πβ².
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 Pβ and Qβ is exactly the length of ππ
multiplied by the scale factor.
2. The lines containing the segments PβQβ and PQ are parallel or equal.
For example, if points P and Q are dilated from center O by a scale factor of π = !!, then the lines
containing the segments PβQβ and PQ are parallel, and πβπβ = !!ππ, as shown below.
πβπβ =32ππ
πβπβ =32 5 =
152 = 7.5
MOD2 L5 3
PRACTICE
1. Produce a scale drawing of βπ·πΊπ» 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 βπ·πΈπΉ is a scale drawing of βπ·πΊπ».
Explain why or why not? Is πΈπΉ β₯ πΊπ»?
π·πΈ = 3.2ππ, πΈπΊ = 3.75ππ, πΈπΉ = 5.9ππ, πΊπ» = 11.9ππ
π·πΈ = ππ·πΊ
3.2 = π(3.2+ 3.75)
π =3.26.95 = 0.46
πΈπΉ = ππΊπ»
5.9 = π(11.9)
π =5.911.9 = 0.496
The scale factors are not the same. Therefore βπ·πΈπΉ is not a scale drawing of βπ·πΊπ» and πΈπΉ β¦ πΊπ».
MOD2 L5 4
3. βπ΄π΅β²πΆβ² is a dilation of βπ΄π΅πΆ from vertex A, and πΆπΆβ = 2. Use the given information and the
diagram to find π΅β²πΆβ².
β’ π΄πΆ = 4 and π΅πΆ = 7
β’ ON YOUR OWN: π΄πΆ = 7 and π΅πΆ = 9
π΄πΆβ² = ππ΄πΆ
(4+ 2) = π(4)
π =64 =
32 = 1.5
π΅πΆβ² = ππ΅πΆ
π΅πΆ! =32 7 ππ (1.5)(7)
π΅πΆ! =212 ππ 10.5
π΄πΆβ² = ππ΄πΆ
(7+ 2) = π(7)
π =97 = 1.286
π΅πΆβ² = ππ΅πΆ
π΅πΆ! =97 9 ππ (1.286)(9)
π΅πΆ! = 817 ππ 11.574
SUMMARY
β’ Dilation Theorem: If a dilation with center O and scale factor r sends point P to Pβ and Q to Qβ,
then πβ²πβ² = π ππ . Furthermore, if π β 1 and O, P, and Q are vertices of a triangle, then
ππ β₯ πβ²πβ².
β’ Three methods for scale drawings:
o Compass Construction
o Ratio Method (Using Dilation)
o Parallel Method
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