Question

determining if figures are similar and related by a sequence of
(a) which sequence(s) of transformations will map figure a
onto figure b exactly? choose all that apply.
reflect figure a over the x - axis, and then dilate that
result with scale factor \\( \frac { 1 } { 2 } \\) centered at the origin.
rotate figure a counterclockwise \\( 180 ^ { \circ } \\) about the origin,
and then dilate that result with scale factor \\( \frac { 1 } { 2 } \\) centered
at the origin.
dilate figure a with scale factor \\( \frac { 1 } { 2 } \\) centered at the
origin, and then reflect that result over the x - axis.
dilate figure a with scale factor \\( \frac { 1 } { 2 } \\) centered at the
origin, and then rotate that result clockwise \\( 90 ^ { \circ } \\) about
the origin.
none of these
(b) are figure a and figure b similar?

Explanation:

Step1: Analyze each transformation sequence

• First option: Reflecting over the \(x -\)axis changes the \(y -\)coordinates' sign. Then dilating with scale factor \(\frac{1}{2}\) centered at the origin. But the orientation and position won't match.
• Second option: Rotating \(180^{\circ}\) counter - clockwise about the origin changes \((x,y)\) to \((-x,-y)\). Then dilating with scale factor \(\frac{1}{2}\) centered at the origin. The orientation and size won't match.
• Third option: Dilating with scale factor \(\frac{1}{2}\) centered at the origin first. Then reflecting over the \(x -\)axis. The orientation and position won't match.
• Fourth option: Dilate Figure A with scale factor \(\frac{1}{2}\) centered at the origin. Then rotate the result clockwise \(90^{\circ}\) about the origin.

Let's assume a general point \((x,y)\) on Figure A. After dilation: \((\frac{1}{2}x,\frac{1}{2}y)\). After a \(90^{\circ}\) clockwise rotation about the origin (the transformation rule \((a,b)\to(b, - a)\)): \((\frac{1}{2}y,-\frac{1}{2}x)\) which can map Figure A to Figure B.

Step2: Determine similarity

Since there is a sequence of transformations (dilation and rotation) that can map Figure A to Figure B, and dilation is a similarity transformation (preserves shape, changes size proportionally) and rotation is a rigid transformation (preserves shape and size).

Answer:

(a) Dilate Figure A with scale factor \(\frac{1}{2}\) centered at the origin, and then rotate that result clockwise \(90^{\circ}\) about the origin.
(b) Yes, Figure A and Figure B are similar.