Question

quadrilateral

ed by a scale factor of \\( \frac { 1 } { 2 } \\) with point \\( r \\) as the center of dilation, resulting in the image \\( p ^ { \prime } q ^ { \prime } r ^ { \prime } s ^ { \prime } \\). which statement is true about line segment \\( p ^ { \prime } s ^ { \prime } \\)?

\\( \bigcirc \\) a. segment \\( p ^ { \prime } s ^ { \prime } \\) is 4 units long and lies on a different line than segment \\( p s \\).
\\( \bigcirc \\) b. segment \\( p ^ { \prime } s ^ { \prime } \\) is 4 units long and lies on the same line as segment \\( p s \\).
\\( \bigcirc \\) c. segment \\( p ^ { \prime } s ^ { \prime } \\) is 16 units long and lies on the same line as segment \\( p s \\).
\\( \bigcirc \\) d. segment \\( p ^ { \prime } s ^ { \prime } \\) is 16 units long and lies on a different line than segment \\( p s \\).

Explanation:

Step1: Determine the length of segment PS

From the graph, the coordinates of \(P\) are \((- 8,8)\) and of \(S\) are \((-8,0)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(x_1=-8,y_1 = 8,x_2=-8,y_2 = 0\). So \(PS=\sqrt{(-8+8)^2+(0 - 8)^2}=\sqrt{0+64}=8\) units.

Step2: Calculate the length of segment \(P'S'\)

Since the scale factor \(k=\frac{1}{2}\) and the formula for the length of a dilated segment (when the center of dilation is a vertex) is \(L'=k\times L\). Here \(L = PS = 8\), so \(P'S'=\frac{1}{2}\times8 = 4\) units.

Step3: Analyze the position of \(P'S'\)

Dilation with center \(R\) (a vertex of the quadrilateral) and for a segment like \(PS\) (where \(P\) and \(S\) are vertices of the original quadrilateral), points \(P\) and \(S\) are collinear with their images \(P'\) and \(S'\) (because dilation is a similarity transformation that preserves collinearity when the center of dilation is a common - point related to the figure). So \(P'S'\) lies on the same line as \(PS\)

Answer:

B. Segment \(P'S'\) is 4 units long and lies on the same line as segment \(PS\)