Question

when the center of dilation is at the origin and the scale factor is 0.5, what happens to the figure?
a. the figure enlarges to twice its original size
b. the figure remains the same size
c. the figure reduces to half its original size
d. the figure disappears

a line segment with endpoints p(1,2) and q(3,4) is dilated with a scale factor of ½. what are the coordinates of q’?
a. (6,8)
b. (1.5,2)
c. (2,3)
d. (1.5,3)

how are the coordinates of a dilated point calculated?
a. subtract the center of dilation from the coordinates
b. divide each coordinate by the scale factor
c. multiply each coordinate by the scale factor
d. add the scale factor to each coordinate

what is the result of dilating a figure with a scale factor of 1?
a. the figure is shifted to a new position
b. the figure is enlarged by twice
c. the figure remains the same size
d. the figure is reduced by half

Explanation:

First Question

When the center of dilation is at the origin and the scale factor \(k = 0.5=\frac{1}{2}\), for any point \((x,y)\) on the original figure, the coordinates of the dilated point \((x',y')\) are given by \((x',y')=(kx,ky)\). This means each side - length of the figure is multiplied by \(0.5\). If the original side - length is \(s\), the new side - length is \(s\times0.5=\frac{s}{2}\). So the figure reduces to half its original size.

Second Question

Given a point \(Q(3,4)\) and a scale factor \(k = 2\). When dilating a point \((x,y)\) with a scale factor \(k\) about the origin, the formula for the dilated point \((x',y')\) is \((x',y')=(kx,ky)\). Substitute \(x = 3\) and \(y = 4\) and \(k = 2\) into the formula: \(x'=2\times3 = 6\) and \(y'=2\times4=8\). So the coordinates of \(Q'\) are \((6,8)\).

Third Question

The formula for dilating a point \((x,y)\) with a scale factor \(k\) about the origin is \((x',y')=(kx,ky)\). This means we multiply each coordinate by the scale factor.

Fourth Question

When the scale factor \(k = 1\), for a point \((x,y)\) on the original figure, the dilated point \((x',y')=(kx,ky)=(1\times x,1\times y)=(x,y)\). So each point of the figure maps to itself, and the figure remains the same size.

Answer:

1. C. The figure reduces to half its original size
2. A. \((6,8)\)
3. C. Multiply each coordinate by the scale factor
4. C. The figure remains the same size