Question

b. dilate each figure or answer the question using the information provided. then, compare and contrast the corresponding angles and side lengths of the original figure and dilated figure. in addition, compare the area of the original figure to the area of the dilated figure. finally, write the algebraic representation for a dilation of the point ( x, y) using the given scale factor. 2. dilate △abc on the coordinate plane using the origin (0, 0) as the center of dilation and a scale factor of 1/2. the corresponding angles are all original side length : new side length is the area of the original figure is 32 units² the area of the dilated figure is 32 x = units² original area : new area is the algebraic representation is ( )

Explanation:

Step1: Property of angles in dilation

In a dilation, corresponding angles of the original and dilated figures are congruent.

Step2: Ratio of side - lengths

The scale factor \(k = \frac{1}{2}\). The ratio of the original side - length to the new side - length is \(\frac{1}{k}=2:1\).

Step3: Area relationship

The ratio of the areas of two similar figures is equal to the square of the scale factor. Since \(k=\frac{1}{2}\), the area of the dilated figure \(A_{new}=A_{old}\times k^{2}\), where \(A_{old} = 32\) and \(k=\frac{1}{2}\), so \(A_{new}=32\times(\frac{1}{2})^2=8\).

Step4: Ratio of areas

The ratio of the original area to the new area is \(\frac{A_{old}}{A_{new}}=\frac{32}{8}=4:1\).

Step5: Algebraic representation

When the center of dilation is the origin and the scale factor is \(k=\frac{1}{2}\), the algebraic representation for a point \((x,y)\) is \((kx,ky)=(\frac{1}{2}x,\frac{1}{2}y)\).

Answer:

1. The corresponding angles are all equal.
2. Original Side Length : New Side Length is 2 : 1.
3. The area of the dilated figure is \(32\times(\frac{1}{2})^2 = 8\) units². So the blanks are \((\frac{1}{2})^2\) and \(8\).
4. Original Area : New Area is 4 : 1.
5. The algebraic representation is \((\frac{1}{2}x,\frac{1}{2}y)\).