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NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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Lesson 6: Dilations on the Coordinate Plane
Student Outcomes
Students describe the effect of dilations on two-dimensional figures using coordinates.
Classwork
Example 1 (7 minutes)
Students learn the multiplicative effect of scale factor on a point. Note that this effect holds when the center of dilation
is the origin. In this lesson, the center of any dilation used is always assumed to be (0, 0).
Show the diagram below, and ask students to look at and write or share a claim about the effect that dilation has on the
coordinates of dilated points.
The graph below represents a dilation from center (0, 0) by scale factor ๐ = 2.
Show students the second diagram below so they can check if their claims were correct. Give students time to verify the
claims that they made about the above graph with the one below. Then, have them share their claims with the class.
Use the discussion that follows to crystallize what students observed.
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NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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The graph below represents a dilation from center (0, 0) by scale factor ๐ = 4.
In Lesson 5, we found the location of a dilated point by using the knowledge of dilation and scale factor, as well
as the lines of the coordinate plane to ensure equal angles, to find the coordinates of the dilated point. For
example, we were given the point ๐ด(5, 2) and told the scale factor of dilation was ๐ = 2. Remember that the
center of this dilation is (0,0). We created the following picture and determined the location of ๐ดโฒ to be
(10, 4).
We can use this information and the observations we made at the beginning of class to develop a shortcut for
finding the coordinates of dilated points when the center of dilation is the origin.
Notice that the horizontal distance from the ๐ฆ-axis to point ๐ด was multiplied by a scale factor of 2. That is, the
๐ฅ-coordinate of point ๐ด was multiplied by a scale factor of 2. Similarly, the vertical distance from the ๐ฅ-axis to
point ๐ด was multiplied by a scale factor of 2.
MP.8
MP.3
NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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Here are the coordinates of point ๐ด(5, 2) and the dilated point ๐ดโฒ(10, 4). Since the scale factor was 2, we can
more easily see what happened to the coordinates of ๐ด after the dilation if we write the coordinates of ๐ดโฒ as
(2 โ 5, 2 โ 2), that is, the scale factor of 2 multiplied by each of the coordinates of ๐ด to get ๐ดโฒ.
The reasoning goes back to our understanding of dilation. The length ๐|๐๐ต| = |๐๐ตโฒ|, by the definition of
dilation, and the length ๐|๐ด๐ต| = |๐ดโฒ๐ตโฒ|; therefore,
๐ =|๐๐ตโฒ|
|๐๐ต|=
|๐ดโฒ๐ตโฒ|
|๐ด๐ต|,
where the length of the segment ๐๐ตโฒ is the ๐ฅ-coordinate of the dilated point (i.e., 10), and the length of the
segment ๐ดโฒ๐ตโฒ is the ๐ฆ-coordinate of the dilated point (i.e., 4).
In other words, based on what we know about the lengths of dilated segments, when the center of dilation is
the origin, we can determine the coordinates of a dilated point by multiplying each of the coordinates in the
original point by the scale factor.
Example 2 (3 minutes)
Students learn the multiplicative effect of scale factor on a point.
Letโs look at another example from Lesson 5. We were given the point ๐ด(7, 6) and asked to find the location
of the dilated point ๐ดโฒ when ๐ =117
. Our work on this problem led us to coordinates of approximately
(11, 9.4) for point ๐ดโฒ. Verify that we would get the same result if we multiply each of the coordinates of point
๐ด by the scale factor.
๐ดโฒ (117
โ 7,117
โ 6)
11
7โ 7 = 11
and
11
7โ 6 =
66
7โ 9.4
Therefore, multiplying each coordinate by the scale factor produced the desired result.
Example 3 (5 minutes)
The coordinates in other quadrants of the graph are affected in the same manner as we have just seen. Based
on what we have learned so far, given point ๐ด(โ2, 3), predict the location of ๐ดโฒ when ๐ด is dilated from a
center at the origin, (0, 0), by scale factor ๐ = 3.
Provide students time to predict, justify, and possibly verify, in pairs, that ๐ดโฒ(3 โ (โ2), 3 โ 3) = (โ6, 9). Verify the fact
on the coordinate plane, or have students share their verifications with the class.
MP.8
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Lesson 6: Dilations on the Coordinate Plane
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As before, mark a point ๐ต on the ๐ฅ-axis. Then, |๐๐ตโฒ| = 3|๐๐ต|. Where is point ๐ตโฒ located?
Since the length of |๐๐ต| = 2, then |๐๐ตโฒ| = 3 โ 2 = 6. But we are looking at a distance to the left of
zero; therefore, the location of ๐ตโฒ is (โ6, 0).
Now that we know where ๐ตโฒ is, we can easily find the location of ๐ดโฒ. It is on the ray ๐๐ดโโโโ โ, but at what location?
The location of ๐ดโฒ(โ6, 9), as desired
NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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Exercises 1โ5 (5 minutes)
Students complete Exercises 1โ5 independently.
Exercises 1โ5
1. Point ๐จ(๐, ๐) is dilated from the origin by scale factor ๐ = ๐. What are the coordinates of point ๐จโฒ?
๐จโฒ(๐ โ ๐, ๐ โ ๐) = ๐จโฒ(๐๐, ๐๐)
2. Point ๐ฉ(โ๐, ๐) is dilated from the origin by scale factor ๐ =๐๐
. What are the coordinates of point ๐ฉโฒ?
๐ฉโฒ (๐
๐โ (โ๐),
๐
๐โ ๐) = ๐ฉโฒ (โ๐,
๐
๐)
3. Point ๐ช(๐,โ๐) is dilated from the origin by scale factor ๐ =๐๐
. What are the coordinates of point ๐ชโฒ?
๐ชโฒ (๐
๐โ ๐,
๐
๐โ (โ๐)) = ๐ชโฒ (
๐
๐, โ
๐
๐)
4. Point ๐ซ(๐, ๐๐) is dilated from the origin by scale factor ๐ = ๐. What are the coordinates of point ๐ซโฒ?
๐ซโฒ(๐ โ ๐, ๐ โ ๐๐) = ๐ซโฒ(๐, ๐๐)
5. Point ๐ฌ(โ๐,โ๐) is dilated from the origin by scale factor ๐ =๐๐
. What are the coordinates of point ๐ฌโฒ?
๐ฌโฒ (๐
๐โ (โ๐),
๐
๐โ (โ๐)) = ๐ฌโฒ (โ๐,โ
๐๐
๐)
Example 4 (4 minutes)
Students learn the multiplicative effect of scale factor on a two-dimensional figure.
Now that we know the multiplicative relationship between a point and its dilated location (i.e., if point
๐(๐1, ๐2) is dilated from the origin by scale factor ๐, then ๐โฒ(๐๐1, ๐๐2)), we can quickly find the coordinates of
any point, including those that comprise a two-dimensional figure, under a dilation of any scale factor.
For example, triangle ๐ด๐ต๐ถ has coordinates ๐ด(2, 3), ๐ต(โ3, 4), and ๐ถ(5, 7). The triangle is being dilated from
the origin with scale factor ๐ = 4. What are the coordinates of triangle ๐ดโฒ๐ตโฒ๐ถโฒ?
First, find the coordinates of ๐ดโฒ.
๐ดโฒ(4 โ 2, 4 โ 3) = ๐ดโฒ(8, 12)
Next, locate the coordinates of ๐ตโฒ.
๐ตโฒ(4 โ (โ3), 4 โ 4) = ๐ตโฒ(โ12, 16)
Finally, locate the coordinates of ๐ถโฒ.
๐ถโฒ(4 โ 5, 4 โ 7) = ๐ถโฒ(20, 28)
Therefore, the vertices of triangle ๐ดโฒ๐ตโฒ๐ถโฒ have coordinates of (8, 12), (โ12, 16), and (20, 28), respectively.
NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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Example 5 (4 minutes)
Students learn the multiplicative effect of scale factor on a two-dimensional figure.
Parallelogram ๐ด๐ต๐ถ๐ท has coordinates of (โ2, 4), (4, 4), (2, โ1), and (โ4,โ1), respectively. Find the
coordinates of parallelogram ๐ดโฒ๐ตโฒ๐ถโฒ๐ทโฒ after a dilation from the origin with a scale factor ๐ =12
.
๐ดโฒ (12โ (โ2),
12โ 4) = ๐ดโฒ(โ1, 2)
๐ตโฒ (12โ 4,
12โ 4) = ๐ตโฒ(2, 2)
๐ถโฒ (12โ 2,
12โ (โ1)) = ๐ถโฒ (1, โ
12)
๐ทโฒ (12โ (โ4),
12โ (โ1)) = ๐ทโฒ (โ2,โ
12)
Therefore, the vertices of parallelogram ๐ดโฒ๐ตโฒ๐ถโฒ๐ทโฒ have coordinates of (โ1, 2), (2, 2), (1, โ12), and (โ2,โ
12),
respectively.
Exercises 6โ8 (9 minutes)
Students complete Exercises 6โ8 independently.
Exercises 6โ8
6. The coordinates of triangle ๐จ๐ฉ๐ช are shown on the coordinate plane below. The triangle is dilated from the origin by
scale factor ๐ = ๐๐. Identify the coordinates of the dilated triangle ๐จโฒ๐ฉโฒ๐ชโฒ.
Point ๐จ(โ๐, ๐), so ๐จโฒ(๐๐ โ (โ๐), ๐๐ โ ๐) = ๐จโฒ(โ๐๐, ๐๐).
Point ๐ฉ(โ๐,โ๐), so ๐ฉโฒ(๐๐ โ (โ๐), ๐๐ โ (โ๐)) = ๐ฉโฒ(โ๐๐,โ๐๐).
Point ๐ช(๐, ๐), so ๐ชโฒ(๐๐ โ ๐, ๐๐ โ ๐) = ๐ชโฒ(๐๐, ๐๐).
The coordinates of the vertices of triangle ๐จโฒ๐ฉโฒ๐ชโฒ are (โ๐๐, ๐๐), (โ๐๐,โ๐๐), and (๐๐, ๐๐), respectively.
NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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7. Figure ๐ซ๐ฌ๐ญ๐ฎ is shown on the coordinate plane below. The figure is dilated from the origin by scale factor ๐ =๐๐
.
Identify the coordinates of the dilated figure ๐ซโฒ๐ฌโฒ๐ญโฒ๐ฎโฒ, and then draw and label figure ๐ซโฒ๐ฌโฒ๐ญโฒ๐ฎโฒ on the coordinate
plane.
Point ๐ซ(โ๐, ๐), so ๐ซโฒ (๐๐
โ (โ๐),๐๐
โ ๐) = ๐ซโฒ(โ๐, ๐).
Point ๐ฌ(โ๐,โ๐), so ๐ฌโฒ (๐๐
โ (โ๐),๐๐
โ (โ๐)) = ๐ฌโฒ (โ๐๐, โ๐).
Point ๐ญ(๐,โ๐), so ๐ญโฒ (๐๐
โ ๐,๐๐
โ (โ๐)) = ๐ญโฒ (๐๐๐
, โ๐๐ ). -
Point ๐ฎ(โ๐, ๐), so ๐ฎโฒ (๐๐โ (โ๐),
๐๐โ ๐) = ๐ฎโฒ(โ๐, ๐).
The coordinates of the vertices of figure ๐ซโฒ๐ฌโฒ๐ญโฒ๐ฎโฒ are (โ๐, ๐), (โ๐๐, โ๐), (
๐๐๐
, โ๐๐), and (โ๐, ๐), respectively.
NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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8. The triangle ๐จ๐ฉ๐ช has coordinates ๐จ(๐, ๐), ๐ฉ(๐๐, ๐), and ๐ช(๐, ๐๐). Draw and label triangle ๐จ๐ฉ๐ช on the coordinate
plane. The triangle is dilated from the origin by scale factor ๐ =๐๐
. Identify the coordinates of the dilated triangle
๐จโฒ๐ฉโฒ๐ชโฒ, and then draw and label triangle ๐จโฒ๐ฉโฒ๐ชโฒ on the ๐
๐ coordinate plane.
Point ๐จ(๐, ๐), then ๐จโฒ (๐๐
โ ๐,๐๐
โ ๐) = ๐จโฒ(๐,๐๐).
Point ๐ฉ(๐๐, ๐), so ๐ฉโฒ (๐๐โ ๐๐,
๐๐โ ๐) = ๐ฉโฒ(๐, ๐).
Point ๐ช(๐, ๐๐), so ๐ชโฒ (๐๐โ ๐,
๐๐โ ๐๐) = ๐ชโฒ(๐, ๐).
The coordinates of triangle ๐จโฒ๐ฉโฒ๐ชโฒ are (๐,๐๐), (๐, ๐), and (๐, ๐), respectively.
NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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Closing (4 minutes)
Summarize, or ask students to summarize, the main points from the lesson.
We know that we can calculate the coordinates of a dilated point given the coordinates of the original point
and the scale factor.
To find the coordinates of a dilated point, we must multiply both the ๐ฅ-coordinate and the ๐ฆ-coordinate by the
scale factor of dilation.
If we know how to find the coordinates of a dilated point, we can find the location of a dilated triangle or other
two-dimensional figure.
Exit Ticket (4 minutes)
Lesson Summary
Dilation has a multiplicative effect on the coordinates of a point in the plane. Given a point (๐, ๐) in the plane, a
dilation from the origin with scale factor ๐ moves the point (๐, ๐) to (๐๐, ๐๐).
For example, if a point (๐,โ๐) in the plane is dilated from the origin by a scale factor of ๐ = ๐, then the coordinates
of the dilated point are (๐ โ ๐, ๐ โ (โ๐)) = (๐๐,โ๐๐).
NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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Name Date
Lesson 6: Dilations on the Coordinate Plane
Exit Ticket
1. The point ๐ด(7, 4) is dilated from the origin by a scale factor ๐ = 3. What are the coordinates of point ๐ดโฒ?
2. The triangle ๐ด๐ต๐ถ, shown on the coordinate plane below, is dilated from the origin by scale factor ๐ =12
. What is the
location of triangle ๐ดโฒ๐ตโฒ๐ถโฒ? Draw and label it on the coordinate plane.
NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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Exit Ticket Sample Solutions
1. The point ๐จ(๐, ๐) is dilated from the origin by a scale factor ๐ = ๐. What are the coordinates of point ๐จโฒ?
Since point ๐จ(๐, ๐), then ๐จโฒ(๐ โ ๐, ๐ โ ๐) = ๐จโฒ(๐๐, ๐๐).
2. The triangle ๐จ๐ฉ๐ช, shown on the coordinate plane below, is dilated from the origin by scale factor ๐ =๐๐
. What is
the location of triangle ๐จโฒ๐ฉโฒ๐ชโฒ? Draw and label it on the coordinate plane.
Point ๐จ(๐, ๐), so ๐จโฒ (๐๐โ ๐,
๐๐โ ๐) = ๐จโฒ (
๐๐, ๐).
Point ๐ฉ(โ๐, ๐), so ๐ฉโฒ (๐๐โ (โ๐),
๐๐โ ๐) = ๐ฉโฒ (โ
๐๐, ๐).
Point ๐ช(๐, ๐), so ๐ชโฒ (๐๐โ ๐,
๐๐โ ๐) = ๐ชโฒ(๐, ๐).
The coordinates of the vertices of triangle ๐จโฒ๐ฉโฒ๐ชโฒ are (๐๐, ๐), (โ
๐๐, ๐), and (๐, ๐), respectively.
NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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Problem Set Sample Solutions
Students practice finding the coordinates of dilated points of two-dimensional figures.
1. Triangle ๐จ๐ฉ๐ช is shown on the coordinate plane below. The triangle is dilated from the origin by scale factor ๐ = ๐.
Identify the coordinates of the dilated triangle ๐จโฒ๐ฉโฒ๐ชโฒ.
Point ๐จ(โ๐๐, ๐), so ๐จโฒ(๐ โ (โ๐๐), ๐ โ ๐) = ๐จโฒ(โ๐๐, ๐๐).
Point ๐ฉ(โ๐๐, ๐), so ๐ฉโฒ(๐ โ (โ๐๐), ๐ โ ๐) = ๐ฉโฒ(โ๐๐, ๐).
Point ๐ช(โ๐, ๐), so ๐ชโฒ(๐ โ (โ๐), ๐ โ ๐) = ๐ชโฒ(โ๐๐, ๐๐).
The coordinates of the vertices of triangle ๐จโฒ๐ฉโฒ๐ชโฒ are (โ๐๐, ๐๐), (โ๐๐, ๐), and (โ๐๐, ๐๐), respectively.
NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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2. Triangle ๐จ๐ฉ๐ช is shown on the coordinate plane below. The triangle is dilated from the origin by scale factor ๐ =๐๐
.
Identify the coordinates of the dilated triangle ๐จโฒ๐ฉโฒ๐ชโฒ.
Point ๐จ(โ๐๐,โ๐), so ๐จโฒ (๐๐โ (โ๐๐),
๐๐โ (โ๐)) = ๐จโฒ (โ
๐๐๐
, โ๐๐).
Point ๐ฉ(โ๐๐,โ๐), so ๐ฉโฒ (๐๐โ (โ๐๐),
๐๐โ (โ๐)) = ๐ฉโฒ (โ๐๐,โ
๐๐).
Point ๐ช(โ๐,โ๐), so ๐ชโฒ (๐๐โ (โ๐),
๐๐โ (โ๐)) = ๐ฉโฒ (โ๐,โ
๐๐).
The coordinates of the vertices of triangle ๐จโฒ๐ฉโฒ๐ชโฒ are (โ๐๐๐
, โ๐๐), (โ๐๐,โ๐๐), and (โ๐,โ
๐๐), respectively.
3. The triangle ๐จ๐ฉ๐ช has coordinates ๐จ(๐, ๐), ๐ฉ(๐๐, ๐), and ๐ช(โ๐,๐). The triangle is dilated from the origin by a scale
factor ๐ =๐๐
. Identify the coordinates of the dilated triangle ๐จโฒ๐ฉโฒ๐ชโฒ.
Point ๐จ(๐, ๐), so ๐จโฒ (๐๐โ ๐,
๐๐โ ๐) = ๐จโฒ (๐,
๐๐).
Point ๐ฉ(๐๐, ๐), so ๐ฉโฒ (๐๐โ ๐๐,
๐๐โ ๐) = ๐ฉโฒ(๐, ๐).
Point ๐ช(โ๐, ๐), so ๐ชโฒ (๐๐โ (โ๐),
๐๐โ ๐) = ๐ชโฒ(โ๐, ๐).
The coordinates of the vertices of triangle ๐จโฒ๐ฉโฒ๐ชโฒ are (๐,๐๐), (๐, ๐), and (โ๐, ๐), respectively.
NYS COMMON CORE MATHEMATICS CURRICULUM 8โข3 Lesson 6
Lesson 6: Dilations on the Coordinate Plane
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4. Figure ๐ซ๐ฌ๐ญ๐ฎ is shown on the coordinate plane below. The figure is dilated from the origin by scale factor ๐ =๐๐
.
Identify the coordinates of the dilated figure ๐ซโฒ๐ฌโฒ๐ญโฒ๐ฎโฒ, and then draw and label figure ๐ซโฒ๐ฌโฒ๐ญโฒ๐ฎโฒ on the coordinate
plane.
Point ๐ซ(โ๐, ๐), so ๐ซโฒ (๐๐โ (โ๐),
๐๐โ ๐) = ๐ซโฒ (โ
๐๐,๐๐).
Point ๐ฌ(โ๐,โ๐), so ๐ฌโฒ (๐๐โ (โ๐),
๐๐โ (โ๐)) = ๐ฌโฒ (โ
๐๐, โ
๐๐).
Point ๐ญ(๐, ๐), so ๐ญโฒ (๐๐โ ๐,
๐๐โ ๐) = ๐ญโฒ (๐,
๐๐).
Point ๐ฎ(๐, ๐), so ๐ฎโฒ (๐๐โ ๐,
๐๐โ ๐) = ๐ฎโฒ (๐,
๐๐๐
).
The coordinates of the vertices of figure ๐ซโฒ๐ฌโฒ๐ญโฒ๐ฎโฒ are (โ๐๐,๐๐), (โ
๐๐, โ
๐๐), (๐,
๐๐), and (๐,
๐๐๐
), respectively.
5. Figure ๐ซ๐ฌ๐ญ๐ฎ has coordinates ๐ซ(๐, ๐), ๐ฌ(๐, ๐), ๐ญ(๐,โ๐), and ๐ฎ(โ๐,โ๐). The figure is dilated from the origin by
scale factor ๐ = ๐. Identify the coordinates of the dilated figure ๐ซโฒ๐ฌโฒ๐ญโฒ๐ฎโฒ.
Point ๐ซ(๐, ๐), so ๐ซโฒ(๐ โ ๐, ๐ โ ๐) = ๐ซโฒ(๐, ๐).
Point ๐ฌ(๐, ๐), so ๐ฌโฒ(๐ โ ๐, ๐ โ ๐) = ๐ฌโฒ(๐๐, ๐๐).
Point ๐ญ(๐,โ๐), so ๐ญโฒ(๐ โ ๐, ๐ โ (โ๐)) = ๐ญโฒ(๐๐,โ๐๐).
Point ๐ฎ(โ๐,โ๐), so ๐ฎโฒ(๐ โ (โ๐), ๐ โ (โ๐)) = ๐ฎโฒ(โ๐,โ๐๐).
The coordinates of the vertices of figure ๐ซโฒ๐ฌโฒ๐ญโฒ๐ฎโฒ are (๐, ๐), (๐๐, ๐๐),(๐๐,โ๐๐), and (โ๐,โ๐๐), respectively.