Introduction to Rotation

Q.

A triangle is rotated $90$ degrees about the origin.

Which rule describes the transformation?

Q. 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?

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

Q. Which of the following sequences of transformations has been applied to pentagon $ABCDE$ to obtain $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }{E}^{\prime }$ in the figure given below? Is the transformed image congruent to the preimage?

1. Translation by $2$ units upward and then translation by $3$ units toward the right, no
2. Counterclockwise rotation of ${180}^o$ around $(-3,0)$ and then translation of $2$ units to the right, yes
3. ${180}^o$ counterclockwise rotation around the origin, yes
4. ${90}^o$ clockwise rotation about the point $(1,1)$, yes

Q. The length and width of rectangle $ABCD$ are $10$ inches and $5$ inches, respectively. The rectangle is rotated ${45}^o$ in the clockwise direction to obtain rectangle $EFGH$.


Which of the given statements about rectangles $ABCD$ and $EFGH$ are true?

Statement $1$: The ratio of side $BC$ to the area of rectangle $ABCD$ is equal to the ratio of side $FG$ to the area of rectangle $EFGH$.
Statement $2$: Perimeter of rectangle $EFGH$ = $30$ in

1. Neither statement $1$ nor $2$
2. Statement $1$
3. Statement $2$
4. Both statements $1$ and $2$

Q. Find the image of rectangle $ABCD$ when rotated $180°$ counterclockwise about the origin.
Note: Vertex $A^{\prime },B^{\prime },C^{\prime }$, and $D^{\prime }$ correspond to the image of vertex $A,B,C,$ and $D$, respectively.

1. 
2. 
3. 
4. 

Q. On observing the identical trapezoids ABCD and A'B'C'D', your friend claims that the trapezoid A'B'C'D' is obtained by the translation of the first trapezoid, i.e., ABCD. Is your friend correct?

1. No, it is a reflection, as trapezoid ABCD is reflected along the x-axis.
2. Yes, it is a translation, as trapezoid ABCD is moved 4 units right and 2 units down.
   
3. No, it is a rotation, as trapezoid ABCD is rotated ${180}^{\circ }$ about the origin in the clockwise direction.
   
4. Yes, it is a translation, as trapezoid ABCD is moved 4 units down and 2 units right.
   

Q. Which of the following transformations will produce the image overlapping the pre-image?

1. Clockwise rotation of ${90}^o$ and again clockwise rotation by ${90}^o$
2. Clockwise rotation of ${90}^o$ and then counterclockwise rotation of ${90}^o$
3. Translation by 5 units toward the left and then upward translation by 5 units
4. Translation by 3 units toward the left and then counterclockwise rotation of ${90}^o$

Q. The length and width of rectangle $ABCD$ are $10$ inches and $5$ inches, respectively. The rectangle is rotated ${45}^{\circ }$ in the clockwise direction to obtain rectangle $EFGH$.


Which of the given statements is/are true about the rectangles $ABCD\text{and}EFGH?$

Statement 1: Ratio of side $BC$ to area of rectangle $ABCD$ is equal to the ratio of side $FG$ to the area of the rectangle $EFGH.$
Statement 2: Perimeter of rectangle $EFGH=30$ inches

1. $\text{Both statement 1 and 2}$
2. $\text{Neither statement 1 nor 2}$
3. $\text{Statement 1}$
4. $\text{Statement 2}$

Q.

The vertices of a triangle are $\mathrm{A}(-4,2),\mathrm{B}(-2,2)$ and $\mathrm{C}(-3,4)$. Rotate the triangle as described. Find the coordinates of the image.

$270°$ clockwise about the origin.

Q. Point A is rotated along the circle to reach point A'.

Which of the following rotations can help us achieve it?

1. ${270}^{\circ }$ counterclockwise rotation
2. Neither (A) nor (B)
3. Both (A) and (B)
4. ${90}^{\circ }$ clockwise rotation

Q. In which direction and by what angle should rectangle $ABCD$ be rotated around the point $(0,0)$ such that image $EFGH$ is obtained?

1. Counterclockwise, ${45}^o$
2. Clockwise, ${135}^o$
3. Counterclockwise, ${135}^o$
4. Clockwise, ${45}^o$
   

Q. Which of the following sequence of transformations has been applied on pentagon $ABCDE$ to obtain $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }{E}^{\prime }$ in the figure given below? Is the transformed image congruent to the preimage?

1. A ${90}^{\circ }$ clockwise rotation about the point $(1,1)$, Yes
2. Translation by $2$units upward and then translation by $3$ units toward the right, No
3. Counterclockwise rotation of ${180}^{\circ }$ around $(-3,0)$ and then translation of $2$ units in the right, Yes
4. A ${180}^{\circ }$ counterclockwise rotation around the origin, Yes

Q. Which of the following options gives the correct relation between the congruency of the figures given below?

1. $△FGH\cong △IFJ\text{and}△AED\cong △ABC$
2. $△GHF\cong △IJF\text{and}△ABC\cong △AED$
3. $△HGF\cong △JFI\text{and}△BAC\cong △DEA$
4. $△FGH\cong △FIJ\text{and}△CBA\cong △EDA$

Q. Face $ABCD$ of a cubical block is transformed to $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$. What type of transformation has taken place?

1. Rotation of 180° clockwise about the origin
2. Rotation of 180° counterclockwise about the origin
3. Either A or B
4. Neither A nor B

Q. The position of the triangular flag changed from 1 to 2. Identify the transformation that happened.

1. The transformation followed here is ${90}^{\circ }$ clockwise rotation about the origin.
   
2. The transformation followed here is reflection along the $y$-axis.
   
3. The transformation followed here is 5 units left and 1 unit up translation.
   
4. The transformation followed here is ${270}^{\circ }$ clockwise rotation about the origin
   

Q. A shape is rotated about the origin. The initial coordinates of a point on the shape, the final coordinates of the same point on the shape, and the degree of rotation by which the shape has been rotated about the origin are mentioned in each option.
Which of the following options contains the correct combination?

1. $\begin{array}{ccc}\text{Initial coordinates}& \text{Final coordinates}& \text{Rotation}\\ (4,0)& (0,4)& {90}^{\circ }\text{counterclockwise rotation}\end{array}$
2. All of the above
3. $\begin{array}{ccc}\text{Initial coordinates}& \text{Final coordinates}& \text{Rotation}\\ (4,0)& (0,4)& {270}^{\circ }\text{clockwise rotation}\end{array}$
   
4. $\begin{array}{ccc}\text{Initial coordinates}& \text{Final coordinates}& \text{Rotation}\\ (4,0)& (-4,0)& {180}^{\circ }\text{counterclockwise rotation}\end{array}$

Q. The position of the triangular flag changed from 1 to 2. Identify the transformation that happened.

Q. What type of transformation can you see for polygon ABCDOE in the graph given below?

1. Rotation around the origin
2. Reflection along x-axis
3. Translation centered around origin
4. None of the above

Q. The given figure has a $\Delta PQR$ which is rotated and transformed to $\Delta P^{\prime }{Q}^{\prime }{R}^{\prime }$. Select the statement that holds true for the given transformation.

1. Both B and C
2. The angle of rotation $={90}^{\circ }$ about the origin and the rotation was done clockwise.
   
3. The angle of rotation $={270}^{\circ }$ about the origin and the rotation was done clockwise.
   
4. The angle of rotation $={90}^{\circ }$ about the origin and the rotation was done counterclockwise.
   

Q. Points A and B are rotated along the circumference of the given circle to get points A' and B' in the following manner:
Point A: ${180}^{\circ }$ clockwise rotation
Point B: ${90}^{\circ }$ counterclockwise rotation
Which of the following options represents the correct coordinates of the points A and B after the given rotations?

Statement 1:
The correct coordinates of points A and B after the given rotations are: A'(−3, −4) and B'(0, 5)
Statement 2: Arcs AB and A'B' have the same length.

1. Only statement 1 is correct.
2. Both the statements are incorrect.
3. Both the statements are correct.
4. Only statement 2 is correct.

Q. You decide to build two pools, one for adults and one for kids. Two blueprints are given below. The first one is of the preimage of the pool for adults, and the second is of the image formed after performing a transformation, consisting of both adults and kids pools.
Identify the transformation performed here.

1. None of the above
   
2. Rotation
   
3. Translation
   
4. Reflection along the y-axis