Question

Rectangle ABCD is given by these coordinates: A(4,1), B(2,3), C(5,6), D(7,4). Select all the transformations that will carry the rectangle onto itself. First option: a rotation 270° counterclockwise, then a rotation 90° counterclockwise Second option: a translation 10 units left, 5 units down, 10 units right, and 5 units up Third option: a reflection across the y-axis, then a rotation 270º counterclockwise about the origin Fourth option: a reflection across the x-axis, then a translation 5 units left, then a reflection across the y-axis, then a translation 5 units right Fifth option: a reflection across the y-axis, then a translation 9 units right

Answer 1

Given the rectangle ABCD, whose vertices are:

$A\mleft(4,1\mright),B\mleft(2,3\mright),C\mleft(5,6\mright),D\mleft(7,4\mright)$

You can apply the transformations shown in each option, in order to determine all the transformations that will carry the figure onto itself:

• First option:

By definition, the rule for a rotation of 270 degrees counterclockwise is:

$(x,y)\rightarrow(y,-x)$

And the rule for a rotation of 90 degrees counterclockwise is:

$(x,y)\to(-y,x)$

Therefore, you can choose any vertex of the rectangle and apply the first rule and then the second rule:

$A\mleft(4,1\mright)\rightarrow A^{\prime}\mleft(1,-4\mright)\rightarrow A^{\doubleprime}\rightarrow(4,1)$

Notice that the coordinates of point A'' are equal to the original point A.

• Second option:

The rule for a translation of 10 units left and 5 units down is:

$(x,y)\rightarrow(x-10,y-5)$

And the rule for a translation of 10 units right, and 5 units up is:

$(x,y)\rightarrow(x+10,y+5)$

Then, applying the transformations indicated, you get:

$\begin{gathered} A\mleft(4,1\mright)\rightarrow A^{\prime}(4-10,1-5)=A^{\prime}(-6,-4) \\ \\ A^{\prime}(-6,-4)\rightarrow A^{\prime\prime}(-6+10,-4+5)=A^{\doubleprime}(4,1) \end{gathered}$

Notice that point A'' and point A are equal.

• Third option:

The rule for a reflection across the y-axis is:

$(x,y)\rightarrow\mleft(-x,y\mright)$

Then, applying that transformation and then applying the rule for a rotation of 270 degrees counterclockwise about the Origin (shown in the explanation of the First option), you get:

$A(4,1)\rightarrow A^{\prime}(-4,1)\rightarrow A^{\doubleprime}(1,4)$

The points A and A'' are different.

• Fourth option:

The rule for a reflection across the x-axis is:

$(x,y)\rightarrow(x,-y)$

For a translation of 5 units left is:

$(x,y)\rightarrow(x-5,y)$

And for a translation of 5 units right:

$(x,y)\rightarrow(x+5,y)$

Therefore, knowing those rules and also knowing the rule for a reflection across the y-axis, you can apply all the transformations indicated in the third option, you get:

$A\mleft(4,1\mright)\rightarrow A^{\prime}\mleft(4,-1\mright)\rightarrow A^{\doubleprime}(4-5,-1)=A^{\doubleprime}(-1,-1)$

$A^{\doubleprime}(-1,-1)\rightarrow A^{\doubleprime}^{\prime}(1,-1)\rightarrow A^{\doubleprime\prime^{\prime}}(6,-1)$

Notice that the points A and A'''' do not have the same coordinates.

• Fifth option:

Knowing the rule for a reflection across the y-axis, and knowing that to translate a point 9 units right you must add 9 to the x-coordinate of the points, you get:

$A\mleft(4,1\mright)\rightarrow A^{\prime}\mleft(-4,1\mright)\rightarrow A^{\doubleprime}(-4+9,1)=A^{\doubleprime}(5,1)$

Notice that the points A and A'' do not have the same coordinates.

Hence, the answers are:

• First option.

,

• Second option.