Download Holt McDougal Geometry 7-6

7-6
Dilations and Similarity in the Coordinate Plane
Objectives
Apply similarity properties in the
coordinate plane.
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
__________ – a transformation that changes the
size of a figure but not its shape.
The preimage and the image are always similar.
__________ - describes how much the figure is
enlarged or reduced.
For a dilation with scale factor k, you can find the
image of a point by multiplying each coordinate by
k: (a, b)  (ka, kb).
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
Helpful Hint
If the scale factor of a dilation is greater than 1
(k > 1), it is an enlargement. If the scale factor
is less than 1 (k < 1), it is a reduction.
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
Example 1A: Computer Graphics Application
Draw the border of the photo after a
dilation with scale factor
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
Example 1A Continued
Step 1 Multiply the vertices of the photo A(0, 0),
B(0, 4), C(3, 4), and D(3, 0) by
Rectangle
ABCD
Holt McDougal Geometry
Rectangle
A’B’C’D’
7-6
Dilations and Similarity in the Coordinate Plane
Example 1A Continued
Step 2 Plot points A’(0, 0), B’(0, 10), C’(7.5, 10),
and D’(7.5, 0).
Draw the rectangle.
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
Example 1B
What if…? Draw the border of the original photo
after a dilation with scale factor
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
Example 1B Continued
Step 1 Multiply the vertices of the photo A(0, 0),
B(0, 4), C(3, 4), and D(3, 0) by
Rectangle
ABCD
Holt McDougal Geometry
Rectangle
A’B’C’D’
7-6
Dilations and Similarity in the Coordinate Plane
Example 1B Continued
Step 2 Plot points A’(0, 0), B’(0, 2), C’(1.5, 2),
and D’(1.5, 0).
Draw the rectangle.
2
0
B’
A’
Holt McDougal Geometry
C’
1.5 D’
7-6
Dilations and Similarity in the Coordinate Plane
Example 2A: Finding Coordinates of Similar Triangle
Given that ∆TUO ~ ∆RSO, find the
coordinates of U and the scale
factor.
Since ∆TUO ~ ∆RSO,
Substitute 12 for RO,
9 for TO, and 16 for OY.
12OU = 144
OU = 12
Holt McDougal Geometry
Cross Products Prop.
Divide both sides by 12.
7-6
Dilations and Similarity in the Coordinate Plane
Example 2A Continued
U lies on the y-axis, so its x-coordinate is 0. Since
OU = 12, its y-coordinate must be 12. The
coordinates of U are (0, 12).
So the scale factor is
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
Example 2B
Given that ∆MON ~ ∆POQ and
coordinates P (–15, 0), M(–10, 0),
and Q(0, –30), find the coordinates
of N and the scale factor.
Since ∆MON ~ ∆POQ,
Substitute 10 for OM,
15 for OP, and 30 for OQ.
15 ON = 300
ON = 20
Holt McDougal Geometry
Cross Products Prop.
Divide both sides by 15.
7-6
Dilations and Similarity in the Coordinate Plane
Example 2B Continued
N lies on the y-axis, so its x-coordinate is 0. Since
ON = 20, its y-coordinate must be –20. The
coordinates of N are (0, –20).
So the scale factor is
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
Example 3A: Proving Triangles Are Similar
Given: E(–2, –6), F(–3, –2), G(2, –2), H(–4, 2),
and J(6, 2).
Prove: ∆EHJ ~ ∆EFG.
Step 1 Plot the points
and draw the triangles.
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
Example 3A Continued
Step 2 Use the Distance Formula to find the side lengths.
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
Example 3A Continued
Step 3 Find the similarity ratio.
=2
Since
=2
and E  E, by the Reflexive Property,
∆EHJ ~ ∆EFG by SAS ~ .
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
Example 3B
Given: R(–2, 0), S(–3, 1), T(0, 1), U(–5, 3), and
V(4, 3).
Prove: ∆RST ~ ∆RUV
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
Example 3B Continued
Step 1 Plot the points and draw the triangles.
Y
5
4
U
V
3
2
X
S
1
-7 -6 -5 -4 -3 -2 -1
R
-1
Holt McDougal Geometry
T
1
2
3
4
5
6
7
7-6
Dilations and Similarity in the Coordinate Plane
Example 3B Continued
Step 2 Use the Distance Formula to find the side lengths.
Holt McDougal Geometry
7-6
Dilations and Similarity in the Coordinate Plane
Example 3B Continued
Step 3 Find the similarity ratio.
Since
and R  R, by the Reflexive
Property, ∆RST ~ ∆RUV by SAS ~ .
Holt McDougal Geometry