TEKS: G9B, G11A The student will formulate and test conjectures about the properties and attributes of polygons. The student will use and extend similarity properties and transformations to explore and justify conjectures about geometric figures.
A dilation is a transformation that changes the size of a figure but not its shape. The preimage and the image are always similar. A scale factor 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).
Helpful Hint
If the scale factor of a dilation is greater than 1 (k > 1), it is an enlargement. The scale factor in fraction form is an improper fraction. If the scale factor is less than 1 (k < 1), it is a reduction. The scale factor in fraction form is a proper fraction.
Example: 1 Draw the border of the photo after a dilation with scale factor
Example: 1 cont. Step 1 Multiply the vertices of the photo A(0, 0), B(0, 4), C(3, 4), and
D(3, 0) by Rectangle
ABCD
Rectangle
A’B’C’D’
Example: 1 cont. Step 2 Plot points A’(0, 0), B’(0, 10), C’(7.5, 10), and D’(7.5, 0).
Draw the rectangle.
Example: 2 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
Cross Products Prop. Divide both sides by 12.
Example: 2 cont. U lies on the y-axis, so its x-coordinate is 0. Since OU = 12, its ycoordinate must be 12. The coordinates of U are (0, 12).
So the scale factor is
Example: 3 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.
Example: 3 cont. Step 2 Use the Distance Formula to find the side lengths.
8.2 4.1
11.4 5.7
Example: 3 cont. Step 3 Find the similarity ratio.
S→ A→
S→
8.2 4.1
=2
∡E ∡E
11.4 5.7 Since
=2
and ∡E ∡E, by the Reflexive Property,
∆EHJ ~ ∆EFG by SAS ~ .
Example: 4 Given: R(–2, 0), S(–3, 1), T(0, 1), U(–5, 3), and V(4, 3). Prove: ∆RST ~ ∆RUV Step 1 Plot the points and draw the triangles. Y 5 4
U
V
3
S
1
R
-7 -6 -5 -4 -3 -2 -1 -1
X
2
T 1
2
3
4
5
6
7
Example: 4 cont. Step 2 Use the Distance Formula to find the side lengths.
1.4
4.2
2.2
6.7
Example: 4 cont. Step 3 Find the similarity ratio.
S→ A→ S→
1.4 4.2 ∡R ∡R
2.2 6.7 Since
≈
and ∡R ∡R, by the Reflexive
Property, ∆RST ~ ∆RUV by SAS ~ .
Example: 5 Graph the image of ∆ABC after a dilation with scale factor Verify that ∆A'B'C' ~ ∆ABC.
Example: 5 cont. Step 1 Multiply each coordinate by vertices of ∆A’B’C’.
to find the coordinates of the
Example: 5 cont. Step 2 Graph ∆A’B’C’.
B’ (2, 4)
A’ (0, 2) C’ (4, 0)
Example: 5 cont. Step 3 Use the Distance Formula to find the side lengths.
4.2
2.8
6.7
4.5
6.7
4.5
Example: 5 cont. Step 4 Find the similarity ratio.
S→
2.8 4.2
S→
4.5 ≈ 6.7 4.5 ≈ 6.7
S→ Since
, ∆ABC ~ ∆A’B’C’ by SSS ~.
Example: 6 Graph the image of ∆MNP after a dilation with scale factor 3. Verify that ∆M 'N 'P ' ~ ∆MNP.
Example: 6 cont. Step 1 Multiply each coordinate by 3 to find the coordinates of the vertices of ∆M’N’P’.
Example: 6 cont. Step 2 Graph ∆M’N’P’.
Y 7 6 5 4 3 2 1 1
2
3
4
5
6
7
X
-7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7
Example: 6 cont. Step 3 Use the Distance Formula to find the side lengths.
2.2
6.7
4.1
12.4
4.2
12.7
Example: 6 cont. Step 4 Find the similarity ratio.
S→
6.7 ≈ 2.2 12.4≈ 4.1
S→
12.7 ≈ 4.2
S→
Since
, ∆MNP ~ ∆M’N’P’ by SSS ~.
