Question

cai uses a computer program to resize photos for the online school newspaper. rectangle defg represents the photo. she drags the top right corner of the rectangle along the dashed line to form the image rectangle defg. the center of dilation is at the origin. the scale factor is 1/2. a. what are the coordinates of the vertices of image rectangle defg? show your work. b. check your answer to problem 3a. show your work.

Explanation:

Step1: Recall dilation formula

If the center of dilation is at the origin \((0,0)\) and the scale - factor is \(k\), and a point \((x,y)\) is dilated, the new point \((x',y')\) has coordinates \((x',y')=(kx,ky)\). Let's assume the coordinates of the vertices of rectangle \(DEFG\) are \(D(x_1,y_1)\), \(E(x_2,y_2)\), \(F(x_3,y_3)\), \(G(x_4,y_4)\). Since the scale factor \(k = \frac{1}{2}\), the coordinates of the vertices of the dilated rectangle \(D'E'F'G'\) will be \(D'( \frac{1}{2}x_1,\frac{1}{2}y_1)\), \(E'( \frac{1}{2}x_2,\frac{1}{2}y_2)\), \(F'( \frac{1}{2}x_3,\frac{1}{2}y_3)\), \(G'( \frac{1}{2}x_4,\frac{1}{2}y_4)\). But since the figure is not given with specific coordinates, let's assume \(D(0,0)\), \(E(10,0)\), \(F(10,6)\), \(G(0,6)\) (a common - sense assumption for a rectangle in the coordinate plane).

Step2: Calculate new coordinates

For \(D(0,0)\): \(D'(\frac{1}{2}\times0,\frac{1}{2}\times0)=(0,0)\)
For \(E(10,0)\): \(E'(\frac{1}{2}\times10,\frac{1}{2}\times0)=(5,0)\)
For \(F(10,6)\): \(F'(\frac{1}{2}\times10,\frac{1}{2}\times6)=(5,3)\)
For \(G(0,6)\): \(G'(\frac{1}{2}\times0,\frac{1}{2}\times6)=(0,3)\)

Step3: Check the answer

To check, we can use the property of similar - figures. The ratio of the corresponding sides of the original rectangle and the dilated rectangle should be equal to the scale factor. The length of \(EF\) in rectangle \(DEFG\) is \(6\) and the length of \(E'F'\) in rectangle \(D'E'F'G'\) is \(3\), and \(\frac{3}{6}=\frac{1}{2}\). The length of \(DE\) in rectangle \(DEFG\) is \(10\) and the length of \(D'E'\) in rectangle \(D'E'F'G'\) is \(5\), and \(\frac{5}{10}=\frac{1}{2}\).

Answer:

If we assume \(D(0,0)\), \(E(10,0)\), \(F(10,6)\), \(G(0,6)\) for rectangle \(DEFG\), then the coordinates of rectangle \(D'E'F'G'\) are \(D'(0,0)\), \(E'(5,0)\), \(F'(5,3)\), \(G'(0,3)\)