Question

dashed - lined figure is a dilation image of the solid - line figure. the labeled point is the center of dilation. tell whether the dilation is an enlargement (e) or a reduction (r). then find the scale factor of the dilation (k).
1.
e/r
k =
2.
e/r
k =
3.
e/r
k =
4.
e/r
k =
identify the scale factor:
5.
k =
6.
k =

Explanation:

Step1: Understand the concept of scale factor

The scale factor \(k\) of a dilation is the ratio of the length of a side of the image to the length of the corresponding side of the original figure. If \(k> 1\), it is an enlargement (E); if \(0 < k<1\), it is a reduction (R).

Step2: Solve for problem 1

For problem 1:
The length of the side of the solid - line figure (original) is \(1\) in and the length of the corresponding side of the dashed - line figure (image) is \(2\) in.
The scale factor \(k=\frac{\text{length of image side}}{\text{length of original side}}=\frac{2}{1} = 2\). Since \(k = 2>1\), it is an enlargement (E).

Step3: Solve for problem 2

For problem 2:
The length of the side of the solid - line figure (original) is \(4\) and the length of the corresponding side of the dashed - line figure (image) is \(8\).
The scale factor \(k=\frac{8}{4}=2\). Since \(k = 2>1\), it is an enlargement (E).

Step4: Solve for problem 3

For problem 3:
The length of the side of the solid - line figure (original) is \(9\) and the length of the corresponding side of the dashed - line figure (image) is \(6\).
The scale factor \(k=\frac{6}{9}=\frac{2}{3}\). Since \(0<\frac{2}{3}<1\), it is a reduction (R).

Step5: Solve for problem 4

For problem 4:
The length of the side of the solid - line figure (original) is \(6\) and the length of the corresponding side of the dashed - line figure (image) is \(3\).
The scale factor \(k = \frac{3}{6}=\frac{1}{2}\). Since \(0<\frac{1}{2}<1\), it is a reduction (R).

Step6: Solve for problem 5

For problem 5:
Count the lengths. Suppose we consider a vertical side. The length of the side of the original triangle (solid - line, smaller) can be counted as \(2\) units (from \((0,0)\) to \((1,2)\) in terms of vertical component) and the length of the corresponding side of the image (dashed - line, larger) is \(4\) units.
The scale factor \(k=\frac{4}{2}=2\).

Step7: Solve for problem 6

For problem 6:
Count the lengths. Consider the vertical segment from \(B\) to \(O\) (original, \(BO = 5\) units) and from \(B'\) to \(O\) (image, \(B'O=2.5\) units).
The scale factor \(k=\frac{2.5}{5}=\frac{1}{2}\)

Answer:

1. E, \(k = 2\)
2. E, \(k = 2\)
3. R, \(k=\frac{2}{3}\)
4. R, \(k=\frac{1}{2}\)
5. \(k = 2\)
6. \(k=\frac{1}{2}\)