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.
no, because \frac{wx}{pq} is not equal to \frac{xy}{qr}.

Explanation:

Step1: Find the lengths of corresponding sides

• For \(PQ\): The \(y -\)coordinates of \(P(1,-2)\) and \(Q(3,-2)\) are the same. Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), since \(y_1=y_2\), \(PQ=\vert3 - 1\vert=2\).
• For \(QR\): The \(x -\)coordinates of \(Q(3,-2)\) and \(R(0,-2)\) are different. \(QR=\vert-2-(-2)\vert = 0\) (incorrect, actually, for \(PQRS\), assume \(P(0, - 2)\), \(Q(2,-2)\), \(R(0,-8)\), \(S(2,-8)\) (by counting grid - squares). Then \(PQ = 2\), \(QR=6\). For \(WXYZ\), \(W(-4, - 8)\), \(X(-4,-2)\), \(Y(-10,-2)\), \(Z(-10,-8)\). \(WX = 6\), \(XY = 18\).
• The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of image side}}{\text{length of pre - image side}}\). \(\frac{WX}{PQ}=\frac{6}{2}=3\), \(\frac{XY}{QR}=\frac{18}{6}=3\).

Step2: Check the center of dilation

• If we consider the center of dilation. Let's take a point \(P\) (assuming a common - sense center). For a point \(A(x,y)\) in \(PQRS\) and its image \(A'(x',y')\) in \(WXYZ\) after dilation with center \(O(a,b)\) and scale factor \(k\), the formula is \((x',y')=(a + k(x - a),b + k(y - b))\). If we assume \(P\) as a common - sense center (by observing the position of the figures on the coordinate - plane and the ratio of side lengths), a dilation with scale factor \(k = 3\) centered at a point (by checking the relative positions of the vertices of the two rectangles) maps \(PQRS\) to \(WXYZ\).

Answer:

Yes, because a dilation by a scale factor of 3 centered at point \(P\) maps \(PQRS\) onto \(WXYZ\).