Mathematics
Grade 10
15 min
Dilations: scale factor and classification
Dilations: scale factor and classification
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1
Introduction & Learning Objectives
Learning Objectives
Define dilation, center of dilation, and scale factor.
Calculate the scale factor of a dilation given a pre-image and its corresponding image.
Determine the coordinates of an image after a dilation centered at the origin.
Classify a dilation as an enlargement or a reduction based on the absolute value of its scale factor.
Prove that a dilation results in a similar figure by verifying the properties of similarity.
Differentiate between rigid transformations (isometries) and non-rigid transformations (dilations).
Ever wondered how a photo can be resized on your phone without getting distorted, or how a projector makes a tiny image fill a huge screen? 📸 That's the power of dilations!
In this tutorial, we will explore dilations, a type of transformatio...
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Key Concepts & Vocabulary
TermDefinitionExample
DilationA transformation that produces an image that is the same shape as the original, but is a different size. It is a non-rigid transformation.Using the zoom feature on a digital map to make a city appear larger or smaller is a dilation.
Center of DilationA fixed point in the plane about which all points are expanded or contracted. All lines connecting corresponding points on the pre-image and image will intersect at this point.If you dilate a triangle centered at the origin (0,0), the origin is the one point that does not move.
Scale Factor (k)The ratio of a length of a side on the image to the corresponding length of a side on the pre-image. It determines the amount of enlargement or reduction.If a 5 cm line segment is dilated to become a 15 cm line segment, the...
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Core Formulas
Scale Factor Formula
k = \frac{\text{image length}}{\text{pre-image length}} = \frac{P'A'}{PA}
Use this formula to find the scale factor (k) when you know the length of a segment in the image (P'A') and its corresponding segment in the pre-image (PA).
Dilation Rule (Centered at Origin)
P(x, y) \rightarrow P'(kx, ky)
To find the coordinates of the image point P' after a dilation centered at the origin, multiply both the x- and y-coordinates of the pre-image point P by the scale factor k.
Classification of Dilations
Given a scale factor k:
1. If |k| > 1, the dilation is an enlargement.
2. If 0 < |k| < 1, the dilation is a reduction.
3. If |k| = 1, the dilation is a congruence transformation (no size change).
Use the absolute valu...
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Challenging
The image of a point P after a dilation with scale factor k = 4 is P'(12, -20). What transformation would map P' back to P?
A.dilation with a scale factor of -4
B.dilation with a scale factor of 1/4
C.translation by (-9, 24)
D.reflection across the y-axis
Challenging
A dilation with a scale factor of k = -1/2 is applied to a figure. Which statement best describes the resulting image?
A.The image is an enlargement and is rotated 90°.
B.The image is an enlargement and is rotated 180°.
C.The image is a reduction and is rotated 180°.
D.The image is a reduction and is not rotated.
Challenging
Point A(2a, b) is dilated by a scale factor of k = 3 centered at the origin. The image A' is (18, 12). What is the value of a + b?
A.7
B.10
C.13
D.30
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