1. Properties of Dilations
Mod 3 LSN 2
Properties of Dilations
Activating Prior Knowledge-
Multiply the following:
𝑟, 𝑓𝑜𝑟 𝑟=2
2. 6𝑟, 𝑓𝑜𝑟 𝑟= 1 2 28 3
𝑟, 𝑓𝑜𝑟 𝑟= 1 3
4. 5𝑟, 𝑓𝑜𝑟 𝑟=4 20 5
Tie to LO

2. Today, we will use a compass and a ruler to perform dilations.
Mod 3 LSN 2
Properties of Dilations
Today,
we will use a compass and a ruler to perform dilations.

3. Properties of Dilations
Mod 3 LSN 2
Properties of Dilations
Concept Development
How will dilations effect lines, segments, and rays?
Take 1 minute to make a list of all the ways a dilation will effect lines, segments and rays and be prepared to share with your partner and the class.
CFU

4. Properties of Dilations
Mod 3 LSN 2
Properties of Dilations
Concept Development
Example 1: Given line 𝐿, we will dilate with a scale factor 𝑟=2 from center 𝑂.
First, let’s select a center 𝑂 off the line 𝐿 and two points 𝑃 and 𝑄 on line 𝐿.
CFU

5. Properties of Dilations
Mod 3 LSN 2
Properties of Dilations
Concept Development
Second, we draw rays from center 𝑶 through each of the points 𝑷 and 𝑸. We want to make sure that the points 𝑶, 𝑷, and 𝑷′ (the dilated 𝑷) lie on the same line (i.e., are collinear). That is what keeps the dilated image “in proportion.”
CFU

6. Properties of Dilations
Mod 3 LSN 2
Properties of Dilations
Next, we use our compass to measure the distance from 𝑶 to 𝑷. Do this by putting the point of the compass on point 𝑶 and adjust the radius of the compass to draw an arc through point 𝑷. Once you have the compass set, move the point of the compass to 𝑷 and make a mark along the ray 𝑶𝑷 (without changing the radius of the compass) to mark 𝑷 ′ .
CFU

7. Properties of Dilations
Mod 3 LSN 2
Properties of Dilations
Concept Development
Next, we repeat this process to locate 𝑸 ′ .
CFU

8. Properties of Dilations
Mod 3 LSN 2
Properties of Dilations
Concept Development
Finally, connect points 𝑷 ′ and 𝑸 ′ to draw line 𝑳 ′ .
CFU

9. Properties of Dilations
Mod 3 LSN 2
Properties of Dilations
Concept Development
What do you notice about lines 𝑳 and 𝑳 ′ ?
CFU

10. Properties of Dilations Skill Development/Guided Practice
Mod 3 LSN 2
Properties of Dilations
Skill Development/Guided Practice
Do you think line 𝑳 would still be a line under a dilation with scale factor 𝒓=𝟑? Would the dilated line, 𝑳 ′ , still be parallel to 𝑳?
How would you dilate lines 𝑶𝑷 and 𝑶𝑸 with a scale factor 𝒓=𝟑?
Example 2: Dilate the lines 𝑶𝑷 and 𝑶𝑸 with a scale factor 𝒓=𝟑. Label the points on the lines as 𝑷′′ and 𝑸′′ respectively.
CFU

11. Properties of Dilations Skill Development/Guided Practice
Mod 3 LSN 2
Properties of Dilations
Skill Development/Guided Practice
Here is what would happen with scale factor 𝒓=𝟑.
CFU

12. Properties of Dilations Skill Development/Guided Practice
Mod 3 LSN 2
Properties of Dilations
Skill Development/Guided Practice
What would happen if the center 𝑶 were on line 𝑳?
Example 3: Dilate line segments 𝑶𝑷 and 𝑶𝑸 with a scale factor 𝒓=𝟐.
CFU

13. Properties of Dilations Skill Development/Guided Practice
Mod 3 LSN 2
Properties of Dilations
Skill Development/Guided Practice
Here is what the dilations would look like:
CFU

14. Properties of Dilations
Mod 3 LSN 2
Properties of Dilations
Independent Practice
Exercise
Given center 𝑶 and triangle 𝑨𝑩𝑪, dilate the triangle from center 𝑶 with a scale factor 𝒓=𝟑.
Work with a partner to complete the exercise, be sure to answer all parts a – e.
CFU

15. Properties of Dilations
Mod 3 LSN 2
Properties of Dilations
Independent Practice
a) Note that the triangle 𝑨𝑩𝑪 is made up of segments 𝑨𝑩, 𝑩𝑪, and 𝑪𝑨. Were the dilated images of these segments still segments?
Yes, when dilated, the segments were still segments.
b) Measure the length of the segments 𝑨𝑩 and 𝑨 ′ 𝑩 ′ . What do you notice?
The segment 𝑨 ′ 𝑩 ′ was three times the length of segment 𝑨𝑩. This fits with the definition of dilation, that is, 𝑨 ′ 𝑩 ′ =𝒓 𝑨𝑩 .
CFU

16. Properties of Dilations
Mod 3 LSN 2
Properties of Dilations
Independent Practice
c) Verify the claim you made in part (b) by measuring and comparing the lengths of segments 𝑩𝑪 and 𝑩 ′ 𝑪 ′ and segments 𝑪𝑨 and 𝑪 ′ 𝑨 ′ . What does this mean in terms of the segments formed between dilated points?
This means that dilations affect segments in the same way they do points. Specifically, the lengths of segments are dilated according to the scale factor.
CFU

17. Properties of Dilations
Mod 3 LSN 2
Properties of Dilations
Independent Practice
d) Measure ∠𝑨𝑩𝑪 and ∠𝑨′𝑩′𝑪′. What do you notice?
The angles are equal in measure.
e) Verify the claim you made in part (d) by measuring and comparing ∠𝑩𝑪𝑨 and ∠ 𝑩 ′ 𝑪 ′ 𝑨 ′ and ∠𝑪𝑨𝑩 and ∠ 𝑪 ′ 𝑨 ′ 𝑩 ′ . What does that mean in terms of dilations with respect to angles and their degrees?
It means that dilations map angles to angles, and the dilation preserves the measures of the angles.
CFU

18. Closure- What did you learn? Why is it important?
What effect does a scale factor > 1 have on an image? < 1?
Homework: Page S. 10 Problem Set 1 – 5 all.