Question

is pqrs similar to wxyz? justify your answer.
yes, because a dilation by a scale factor of 2 centered at point p maps pqrs onto wxyz.
yes, because a dilation by a scale factor of 3 centered at point p maps pqrs onto wxyz.

Explanation:

Step1: Calculate the distance from \(P\) to \(Q\) and \(W\) to \(X\)

Using the distance formula for vertical lines (since \(x\) - coordinate is same for \(P(3,4)\) and \(Q(3, - 2)\), \(W(3,4)\) and \(X(3,-8)\)). The distance \(d=\vert y_2 - y_1\vert\).
For \(PQ\): \(d_{PQ}=\vert4-( - 2)\vert=\vert4 + 2\vert = 6\).
For \(WX\): \(d_{WX}=\vert4-( - 8)\vert=\vert4 + 8\vert=12\).

Step2: Calculate the scale - factor

The scale factor \(k=\frac{d_{WX}}{d_{PQ}}\).
\(k=\frac{12}{6}=2\).

Answer:

Yes, because a dilation by a scale factor of 2 centered at point \(P\) maps \(PQRS\) onto \(WXYZ\).