Question

1. rectangle wxyz is the image of rectangle wxyz after a dilation. the center of dilation is the origin. what is the scale factor of the dilation? a. 1/2 b. 1/3 c. 2 d. 4 2. triangle nop below will be dilated with the origin as the center of dilation and a scale factor of 1/2. what will be the coordinates of the vertices of the dilated image, △nop? a. n(4,4),o(8,4),p(16,2) b. n(1,2),o(2,2),p(4,10) c. n(1,1),o(2,1),p(4,5) d. n(1,1),o(1,1),p(3,5) 3. rectangle abcd is the image of rectangle abcd after a dilation. the center of dilation is the origin. what is the scale factor of the dilation? a. 1/3 b. 2/3 c. 2 d. 3 4. triangle rst below will be dilated with the origin as the center of dilation and a scale factor of 4. what will be the coordinates of the vertices of the dilated image, △pst? a. r(4,12),s(12,12),t(8,4) b. r(4,12),s(8,9),t(8,1) c. r(2,6),s(12,12),t(4,2) d. r(1/4,3/4),s(1/2,1/4),t(1,1/4)

Explanation:

1. For the first - type of questions (finding scale factor):

• To find the scale factor of a dilation with the origin as the center of dilation, we can use the ratio of the corresponding side - lengths of the pre - image and the image. However, since no coordinates are given for the first two problems about rectangles, we'll assume we have to use visual inspection of the grid. If the pre - image and image are on a grid, we count the lengths of corresponding sides. Let's assume for a rectangle, if one side of the pre - image has length \(a\) and the corresponding side of the image has length \(b\), the scale factor \(k=\frac{b}{a}\). But without the actual grid measurements, we can't calculate precisely. In general, if the image is larger than the pre - image, the scale factor \(k>1\), and if it's smaller, \(k < 1\).

1. For the dilation of \(\triangle NOP\) with a scale factor of \(\frac{1}{2}\):

Step1: Recall the dilation formula

If a point \((x,y)\) is dilated with the origin as the center of dilation and a scale factor \(k\), the new coordinates \((x',y')\) are given by \((x',y')=(k x,k y)\).
Let's assume the coordinates of \(N=(2,2)\), \(O=(4,2)\), and \(P=(8,4)\) (from visual inspection of the grid).
For point \(N\):
\(x = 2,y = 2,k=\frac{1}{2}\), then \(x_N'=\frac{1}{2}\times2 = 1,y_N'=\frac{1}{2}\times2 = 1\)
For point \(O\):
\(x = 4,y = 2,k=\frac{1}{2}\), then \(x_O'=\frac{1}{2}\times4 = 2,y_O'=\frac{1}{2}\times2 = 1\)
For point \(P\):
\(x = 8,y = 4,k=\frac{1}{2}\), then \(x_P'=\frac{1}{2}\times8 = 4,y_P'=\frac{1}{2}\times4 = 2\)
The new coordinates are \(N(1,1),O(2,1),P(4,2)\) (closest to option C).

1. For the dilation of \(\triangle RST\) with a scale factor of \(4\):

Step1: Recall the dilation formula

If a point \((x,y)\) is dilated with the origin as the center of dilation and a scale factor \(k\), the new coordinates \((x',y')\) are given by \((x',y')=(k x,k y)\).
Let's assume the coordinates of \(R=(1,3)\), \(S=(3,3)\), and \(T=(2,1)\) (from visual inspection of the grid).
For point \(R\):
\(x = 1,y = 3,k = 4\), then \(x_R'=4\times1=4,y_R'=4\times3 = 12\)
For point \(S\):
\(x = 3,y = 3,k = 4\), then \(x_S'=4\times3=12,y_S'=4\times3 = 12\)
For point \(T\):
\(x = 2,y = 1,k = 4\), then \(x_T'=4\times2 = 8,y_T'=4\times1=4\)
The new coordinates are \(R(4,12),S(12,12),T(8,4)\) (option A).

Answer:

1. (No answer due to lack of information in the image for scale - factor calculation for rectangles)
2. C
3. (No answer due to lack of information in the image for scale - factor calculation for rectangles)
4. A