Question

Which of the following sequences of transformations will carry triangle A onto triangle B? OA.Reflect triangle A over the y-axis, rotate it 180°, and translate it 3 units up. ОВ. Reflect triangle A over the y-axis and translate it 2 units to the left and 8 units up. OC. Translate triangle A 1 unit up and 1 unit to the left, rotate it 90° clockwise about the origin, and reflect it over the y-axis. OD. Rotate triangle A 90° clockwise about (-1,-4), reflect it over the x-axis, and translate it 3 units to the left.

Answer 1

The transformation we need is:

Translate triangle A 1 unit up and 1 unit to the left, rotate it 90° clockwise about the origin, and reflect it over the y-axis. Therefore the answer is C.

This comes from the following facts.

A translation 1 unit up and 1 unit to the left is given by:

`(x,y)\rightarrow(x-1,y+1)`

Then the original vertexes take the form:

`\begin{gathered} (-3,-1)\rightarrow(-4,0) \\ (-3,-5)\rightarrow(-4,-4) \\ (-1,-5)\rightarrow(-2,-4) \end{gathered}`

Now, we make a rotation of 90° clockwise, this is given by:

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

Then we have:

`\begin{gathered} (-3,-1)\rightarrow(-4,0)\rightarrow(0,4) \\ (-3,-5)\rightarrow(-4,-4)\rightarrow(-4,4) \\ (-1,-5)\rightarrow(-2,-4)\rightarrow(-4,2) \end{gathered}`

Finally we make a reflection over the y-axis, this is given by:

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

Then we have:

`\begin{gathered} (-3,-1)\rightarrow(-4,0)\rightarrow(0,4)\rightarrow(0,4) \\ (-3,-5)\rightarrow(-4,-4)\rightarrow(-4,4)\rightarrow(4,4) \\ (-1,-5)\rightarrow(-2,-4)\rightarrow(-4,2)\rightarrow(4,2) \end{gathered}`

Therefore the whole transformation is:

`\begin{gathered} (-3,-1)\rightarrow(0,4) \\ (-3,-5)\rightarrow(4,4) \\ (-1,-5)\rightarrow(4,2) \end{gathered}`

And we notice that this transformation takes triangle A onto triangle B.