Question

the isosceles trapezoid pqrs is shown.
a dilation image of pqrs with a scale factor of 6 and the center of dilation at the origin is drawn.
which is the ratio of the length of qr to the length of ps?
a. 2:1
b. 4:1
c. 6:1
d. 12:1

Explanation:

Step1: Find the length of \(QR\) and \(PS\) in the original trapezoid

From the graph, the length of \(QR\) (counting the units on the \(x -\)axis) is \(4-(- 2)=6\) units. The length of \(PS\) is \(4 - 0=4\) units.

Step2: Use the property of dilation

When a figure is dilated with a scale factor \(k\), if the original length of a segment is \(l\), the length of the dilated segment \(l'\) is \(l'=k\times l\).
For the dilated trapezoid \(P'Q'R'S'\), if \(k = 6\), the length of \(Q'R'=6\times QR\) and the length of \(P'S'=6\times PS\).
The ratio \(\frac{Q'R'}{P'S'}=\frac{6\times QR}{6\times PS}\) (by the formula \(l' = k\times l\)). Since \(k = 6\) cancels out, \(\frac{Q'R'}{P'S'}=\frac{QR}{PS}\)
Substitute \(QR = 6\) and \(PS = 3\) (from step 1), \(\frac{Q'R'}{P'S'}=\frac{6}{3}=2:1\)

Answer:

A. \(2:1\)