Question

identify the scale factor
determine whether the dilation from figure a to figure b is a reduction or an enlargement and find its scale factor. write an algebraic rule to describe the transformation with the origin as the center of dilation.
1.
2.
k =
k =
3.
4.
a(-10,-1)
a(4,-2)
\frac{a}{a}=
reduction
k = 3/5
|k|

Explanation:

Step1: Recall dilation concept

Dilation is a transformation that changes the size of a figure. If \(|k|> 1\), it's an enlargement; if \(|k|<1\), it's a reduction. The scale - factor \(k\) can be found by comparing the lengths of corresponding sides or the coordinates of corresponding points.

Step2: Analyze Figure 1

Let's assume we can find a pair of corresponding sides. If we consider a horizontal or vertical side of the figures. Suppose a side of Figure A has length \(a\) and the corresponding side of Figure B has length \(b\). The scale - factor \(k=\frac{b}{a}\). By counting the grid units, if a side of Figure A is 4 units and the corresponding side of Figure B is 2 units, then \(k = \frac{2}{4}=\frac{1}{2}\). Since \(|k|=\frac{1}{2}<1\), it's a reduction. The algebraic rule for a dilation with center at the origin is \((x,y)\to(kx,ky)\), so the rule is \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\).

Step3: Analyze Figure 2

Count the lengths of corresponding sides. If a side of Figure A is 3 units and the corresponding side of Figure B is 6 units, then \(k=\frac{6}{3} = 2\). Since \(|k| = 2>1\), it's an enlargement. The algebraic rule is \((x,y)\to(2x,2y)\).

Step4: Analyze Figure 3

If a side of Figure A is 2 units and the corresponding side of Figure B is 4 units, then \(k=\frac{4}{2}=2\). Since \(|k| = 2>1\), it's an enlargement. The algebraic rule is \((x,y)\to(2x,2y)\).

Step5: Analyze Figure 4

As given, for a point \(A(-10,- 10)\) and its image \(A'(4,-4)\), the scale - factor \(k=\frac{4}{-10}=-\frac{2}{5}\). Since \(|k|=\frac{2}{5}<1\), it's a reduction. The algebraic rule is \((x,y)\to(-\frac{2}{5}x,-\frac{2}{5}y)\).

Answer:

1. Reduction, \(k = \frac{1}{2}\), \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)
2. Enlargement, \(k = 2\), \((x,y)\to(2x,2y)\)
3. Enlargement, \(k = 2\), \((x,y)\to(2x,2y)\)
4. Reduction, \(k=-\frac{2}{5}\), \((x,y)\to(-\frac{2}{5}x,-\frac{2}{5}y)\)