Question

math | graded assignment | unit test, part 2 reasoning about similarity and transformations
(score for question 3: _ of 5 points)

1. consider the following dilation.

(a) is the image of the dilation a reduction or an enlargement of the original figure? explain.
(b) what is the scale factor? explain.
answer:

Explanation:

Step1: Compare side - lengths or coordinates

To determine if it's a reduction or enlargement, compare the coordinates of corresponding points. For example, consider point $M(-3,3)$ and its image $M'(-2,2)$. The distance of $M$ from the origin $d_M=\sqrt{(-3)^2 + 3^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}$, and the distance of $M'$ from the origin $d_{M'}=\sqrt{(-2)^2+2^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$. Since $d_{M'}

Step2: Calculate scale factor

The scale factor $k$ for a dilation from a point $(x,y)$ to $(x',y')$ with the origin as the center of dilation is given by $k=\frac{x'}{x}=\frac{y'}{y}$. Using point $M(-3,3)$ and $M'(-2,2)$, $k=\frac{-2}{-3}=\frac{2}{3}$. We can also check with another point, say $S(6,3)$ and $S'(4,2)$. For these points, $k = \frac{4}{6}=\frac{2}{3}$ and $k=\frac{2}{3}$.

Answer:

(a) The image of the dilation is a reduction of the original figure because the distance of each point in the image from the origin is less than the distance of the corresponding point in the original figure from the origin.
(b) The scale factor is $\frac{2}{3}$. This is found by taking the ratio of the $x$ - coordinates (or $y$ - coordinates) of corresponding points (e.g., for $M(-3,3)$ and $M'(-2,2)$, $\frac{-2}{-3}=\frac{2}{3}$) with the origin as the center of dilation.