This lesson gives students more opportunities to practice drawing dilations precisely by reasoning about the definition of a dilation (MP6). Students construct and compare several examples of dilations, and then measure corresponding lengths and distances in the dilated and original figure. Through experimentation, students conjecture that all distances and lengths in the scaled figure are increased or decreased according to the same ratio given by the scale factor.
In this lesson they justify that the distances to the center of the dilation from the scaled figure and from the original figure have the same ratio as the scale factor because of how dilations are defined. They also notice that the ratios of corresponding lengths within the original and scaled figure are the same as the scale factor, but that this fact doesn’t come directly from the definition. In this course, we don’t prove this relationship, instead we leave it as an assertion, justified only by experimentation.
Note on language: In previous courses, students may have learned that a ratio is an association between two or more quantities. However in more advanced work, such as this course, ratio is commonly used as a synonym for quotient (which some refer to as the value of the ratio). This expanded use of the word ratio comes into play in this lesson when students are asked to write fractions representing ratios and notice connections between the fraction and the scale factor. In later lessons we will refer interchangeably to scale factor, ratio of side lengths, and use “ratio” to mean “the quotient of the side lengths.”
- Comprehend that corresponding lengths in the original figure and image are in the same proportion.
- Determine that dilating by a scale factor of $k$ multiplies all lengths by $k$ (orally and in writing).
- Let’s dilate polygons.
- I know that when figures are dilated by a scale factor of $k$, all lengths in the figure are multiplied by $k$.
A dilation with center \(P\) and positive scale factor \(k\) takes a point \(A\) along the ray \(PA\) to another point whose distance is \(k\) times farther away from \(P\) than \(A\) is.
Triangle \(A'B'C'\) is the result of applying a dilation with center \(P\) and scale factor 3 to triangle \(ABC\).
The factor by which every length in an original figure is increased or decreased when you make a scaled copy. For example, if you draw a copy of a figure in which every length is magnified by 2, then you have a scaled copy with a scale factor of 2.