Question

What transformations should figure $1$ undergo to join figure $2$ to form a square?

• A. Reflection on the $y$-axis $\to$ Translation of three units toward the left $\to$ Translation of two units up $\to$ Dilation about the origin with a scale factor of $2$
• B. Rotation of ${90}^o$ clockwise about the origin $\to$ Translation of three units up $\to$ Translation of three units toward the left $\to$ Dilation about the origin with a scale factor of $2$
• C. Dilation about the origin with a scale factor of $2\to$ Translation of two units up $\to$ Translation of three units toward the left
• D. Dilation about the origin with a scale factor of $2\to$ Rotation of 90° counterclockwise about the origin $\to$ Translation of three units toward the left $\to$ Translation of two units up

Answer 1

The correct option is D Dilation about the origin with a scale factor of $2\to$ Rotation of 90° counterclockwise about the origin $\to$ Translation of three units toward the left $\to$ Translation of two units up

Reflection on the $y$-axis:
Coordinates of $A^{\prime }:(0,0),,$
Coordinates of $B^{\prime }:(-1,0),$
Coordinates of $C^{\prime }:(0,1)$

Translation of three units toward left:
Coordinates of $A^{\prime }:(-3,0),$ Coordinates of $B^{\prime }:(-4,0),$ Coordinates of $C^{\prime }:(-3,1)$

Translation of two units up:
Coordinates of $A^{\prime }:(-3,2),$ Coordinates of $B^{\prime }:(-4,2),$Coordinates of $C^{\prime }:(-3,3)$

Dilation about the origin with a scale factor of $2$:
Coordinates of $A^{\prime }:(-6,4),$ Coordinates of $B^{\prime }:(-8,4),$ Coordinates of $C^{\prime }:(-6,6)$
Resultant image:

Hence, option A is incorrect.

Rotation ${90}^{\circ }$ clockwise about the origin:
Coordinates of $A^{\prime }:(0,0),$ Coordinates of $B^{\prime }:(0,-1),$ Coordinates of $C^{\prime }:(1,0)$
Translation of three units up:
Coordinates of $A^{\prime }:(0,3),$ Coordinates of $B^{\prime }:(0,2),$ Coordinates of $C^{\prime }:(1,3)$
Translation of three units toward the left:
Coordinates of $A^{\prime }:(-3,3),$ Coordinates of $B^{\prime }:(-3,2),$ Coordinates of $C^{\prime }:(-2,3)$
Dilation about the origin with a scale factor of $2:$
Coordinates of $A^{\prime }:(-6,6),$ Coordinates of $B^{\prime }:(-6,4),$ Coordinates of $C^{\prime }:(-4,6)$
Resultant image:


Hence, option B is incorrect.

Dilation about the origin with a scale factor of 2:
Coordinates of $A^{\prime }:(0,0),$ Coordinates of $B^{\prime }:(2,0),$ Coordinates of $C^{\prime }:(0,2)$
Translation of two units up:
Coordinates of $A^{\prime }:(0,2),$ Coordinates of $B^{\prime }:(2,2),$ Coordinates of $C^{\prime }:(0,4)$
Translation of three units toward the left:
Coordinates of $A^{\prime }:(-3,2)$, Coordinates of $B^{\prime }:(-1,2),$ Coordinates of $C^{\prime }:(-3,4)$
Resultant image:


Hence, option D is incorrect.


Step 1: Dilation about the origin with a scale factor of 2


Step 2: Rotation ${90}^{\circ }$ counterclockwise about the origin


Step 3: Translation of three units toward the left


Step 4: Translation of two units up


Hence, the correct answer is option C.