Question

Which of the following sequence of transformations correctly relates the triangles $ABC,DEF,\text{and/or}GHI$ in the figure given below? Will the areas of the triangles mentioned be equal?

• A. Clockwise rotation of $△GHI$ by ${90}^{\circ }$ around $(3,0)$ and left translation by $2$ units will give $ABC$, No
• B. Counterclockwise rotation of $△DEF$ by ${180}^{\circ }$ around $(3,0)$ and upward translation by $2$ units will give $GHI$, Yes
• C. Clockwise rotation of $△ABC$ by ${90}^{\circ }$ around $(1,0)$ and right translation by $2$ units will give $DEF$, Yes
• D. Counterclockwise rotation of $△ABC$ by ${90}^{\circ }$ around the origin and right translation by $2$ units will give $GHI$, Yes

Answer 1

The correct option is C Clockwise rotation of $△ABC$ by ${90}^{\circ }$ around $(1,0)$ and right translation by $2$ units will give $DEF$, Yes

Check each of the options one-by-one and see if the sequence of transformations is correct and then check if the area relationship between the mentioned triangles is correct.

In option A: Counterclockwise rotation of $△DEF$ by ${180}^{\circ }$ around $(3,0)$ and upward translation by $2$ units will give $GHI$, Yes

Let’s visualize this sequence of transformation step-by-step:


1. The above figure shows the counterclockwise rotation of $△DEF$ by ${180}^{\circ }$ around $(3,0)$ that produces the image $D^{\prime }{E}^{\prime }{F}^{\prime }$.


2. The above figure shows the upward translation of the rotated image by $2$ units.
Now, the image $D^{\prime \prime }{E}^{\prime \prime }{F}^{\prime \prime }$ obtained in the above figure does not coincide with $△GHI$ asked in the question, which means the given sequence of transformations is incorrect.

In option B: Counterclockwise rotation of $△ABC$ by ${90}^{\circ }$ around the origin and right translation by $2$ units will give $GHI$, Yes
Let’s visualize this sequence of transformations step-by-step:


The above figure shows the counterclockwise rotation of $△ABC$ by ${90}^{\circ }$ around $(0,0)$ that produces the image $A^{\prime }{B}^{\prime }{C}^{\prime }$, and then rightward translation of the rotated image by $2$ units will produce the image $A^{\prime \prime }{B}^{\prime \prime }{C}^{\prime \prime }$.
Now, the image $A^{\prime \prime }{B}^{\prime \prime }{C}^{\prime \prime }$ obtained in the above figure does not coincide with $△GHI$ asked in the question, as $△GHI$ is lying entirely in the fourth quadrant. Whereas, the image obtained above is touching the negative y-axis, which means the given sequence of transformation is incorrect.

In option C: Clockwise rotation of $△ABC$ by ${90}^{\circ }$ around $(1,0)$ and right translation by $2$ units will give $DEF$, Yes
Let’s visualize this sequence of transformations step-by-step:


The above figure shows the clockwise rotation of $△ABC$ by ${90}^{\circ }$ around $(1,0)$ that produces the image $A^{\prime }{B}^{\prime }{C}^{\prime }$, and then rightward translation of the rotated image by $2$ units will produce the image $A^{\prime \prime }{B}^{\prime \prime }{C}^{\prime \prime }$.

Now, the image $A^{\prime \prime }{B}^{\prime \prime }{C}^{\prime \prime }$ obtained in the above figure coincides with $△DEF$ asked in the question, which means the given sequence of transformations is correct.
The two triangles mentioned in this option are $△ABC$ and $△DEF$ for which we have to check if their areas are equal.

The rotation and translation performed above are rigid transformations and rigid transformations always produce congruent images. Therefore, $△ABC$ and $△DEF$ will be congruent and thus will have equal area.
Hence, this is the correct option.

In option D: Clockwise rotation of $△GHI$ by ${90}^{\circ }$ around $(3,0)$ and left translation by $2$ units will give $ABC$, No
Let’s visualize this sequence of transformations step-by-step:

The above figure shows the clockwise rotation of $△GHI$ by ${90}^{\circ }$ around $(3,0)$ that produces the image $G^{\prime }{H}^{\prime }{I}^{\prime }$, and then the left translation of the rotated image by $2$ units will produce the image $G^{\prime \prime }{H}^{\prime \prime }{I}^{\prime \prime }$.
Now, the image $G^{\prime \prime }{H}^{\prime \prime }{I}^{\prime \prime }$ obtained in the above figure does not coincide with $△ABC$ asked in the question, as $△ABC$ is not touching the origin. Whereas, the image obtained above is touching the origin, which means the given sequence of transformations is incorrect.

➡Option C is correct.