Question

is pqr similar to xyz? justify your answer.
yes, because a dilation by a scale factor of 3 centered at the origin followed by a rotation 90° counterclockwise around the origin maps pqr onto xyz.
yes, because a dilation by a scale factor of 3 centered at point q followed by a rotation 90° counterclockwise around the origin maps pqr onto xyz.
no, because ∠p and ∠x do not have the same measure.

Explanation:

Step1: Find coordinates of points

• Coordinates of \(P(1,-2)\), \(Q(3,-3)\), \(R(0,1)\), \(X(6,3)\), \(Y(9,0)\), \(Z(-3,0)\)

Step2: Calculate scale factor for dilation

• For dilation centered at the origin, if \((x,y)\) is dilated to \((kx,ky)\).
• Consider \(R(0,1)\) and \(X(6,3)\). If we assume dilation first. Let the scale factor be \(k\). For \(y - \)coordinate: if \(y = 1\) and \(ky=3\), then \(k = 3\). For \(x-\)coordinate of \(R(0,1)\) dilated by \(k = 3\) gives \((0,3)\).
• After dilation of \(\triangle PQR\) with scale factor \(k = 3\) centered at the origin: \(P(1,-2)\to(3,-6)\), \(Q(3,-3)\to(9,-9)\), \(R(0,1)\to(0,3)\)
• Then use the rotation formula \((x,y)\to(-y,x)\) (rotation of \(90^{\circ}\) counter - clockwise about the origin).
• For \((3,-6)\) after rotation: \((6,3)\) (matches \(X\)), for \((9,-9)\) after rotation: \((9,9)\) (not relevant as we check correspondence), for \((0,3)\) after rotation: \((- 3,0)\) (matches \(Z\))
• If dilation is centered at \(Q\), the transformation rules will not map the points correctly. Also, if we check angles, since dilation and rotation (both rigid - motion related, dilation preserves angle measures and rotation is a rigid motion)
• Since a dilation (which is a similarity transformation) by scale factor \(k = 3\) centered at the origin followed by a \(90^{\circ}\) counter - clockwise rotation (a rigid motion, which also preserves similarity) maps \(\triangle PQR\) to \(\triangle XYZ\)

Answer:

Yes, because a dilation by a scale factor of 3 centered at the origin followed by a rotation \(90^{\circ}\) counterclockwise around the origin maps \(PQR\) onto \(XYZ\).