Question

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answer the questions about figure a and figure b below.
(a) which sequence(s) of transformations will map figure a onto figure b exactly? choose all that apply.
translate figure a to the right 5 units, and then dilate that result with scale factor 2 centered at the origin.
rotate figure a counterclockwise 180° about the origin, and then dilate that result with scale factor 2 centered at the origin.
dilate figure a with scale factor 2 centered at the origin, and then reflect that result over the y - axis.
dilate figure a with scale factor 2 centered at the origin, and then rotate that result clockwise 90° about the origin.
none of these
(b) are figure a and figure b similar?
yes no

Explanation:

Step1: Analyze the first option

• Translation and dilation:
• If we translate Figure A to the right 5 units, its position changes horizontally. But Figure A is a right - angled triangle and Figure B is a trapezoid. A translation followed by a dilation (which is a similarity transformation) cannot change the shape of a triangle to a trapezoid. So, "Translate Figure A to the right 5 units, and then dilate that result with scale factor 2 centered at the origin" is incorrect.

Step2: Analyze the second option

• Rotation and dilation:
• Rotating Figure A counter - clockwise \(180^{\circ}\) about the origin changes its orientation. But again, a rotation (a rigid transformation) followed by a dilation (a similarity transformation) cannot change the shape of a triangle to a trapezoid. So, "Rotate Figure A counterclockwise \(180^{\circ}\) about the origin, and then dilate that result with scale factor 2 centered at the origin" is incorrect.

Step3: Analyze the third option

• Dilation and reflection:
• Dilation is a similarity transformation. Reflecting over the \(y\) - axis is a rigid transformation. But Figure A is a triangle and Figure B is a trapezoid. A dilation (which scales the figure) followed by a reflection (which flips the figure) cannot change the shape of a triangle to a trapezoid. So, "Dilate Figure A with scale factor 2 centered at the origin, and then reflect that result over the \(y\) - axis" is incorrect.

Step4: Analyze the fourth option

• Dilation and rotation:
• Dilation is a similarity transformation. Rotating clockwise \(90^{\circ}\) about the origin is a rigid transformation. But Figure A is a triangle and Figure B is a trapezoid. A dilation (which scales the figure) followed by a rotation (which turns the figure) cannot change the shape of a triangle to a trapezoid. So, "Dilate Figure A with scale factor 2 centered at the origin, and then rotate that result clockwise \(90^{\circ}\) about the origin" is incorrect.

Step5: Determine similarity

• Similarity criterion:
• Two figures are similar if one can be obtained from the other by a sequence of translations, rotations, reflections, and dilations. Since no sequence of the given transformations (which are translations, rotations, reflections, and dilations) can map Figure A (a triangle) to Figure B (a trapezoid), they are not similar.

Answer:

(a) None of these; (b) No