Question

Perform the following transformations :First reflect the ABC across the y-axis ,then dilate by a factor of 1/2

Answer 1

Given the triangle ABC, you have to reflect it over the y-axis and then dilate it by scale factor k=1/2

- The reflection over the y-axis is a rigid transformation (the figure changes position but it does not change shape), which means that the resulting image will be congruent to the original.

- The dilation is a nonrigid transformation, the figure changes its shape, the resulting image after dilation is similar to the original one.

Reflection over the y-axis ΔABC to ΔA'B'C'

To reflect an image over the y-axis you have to change the sign of the x-coordinate leaving the y-coordinate of each vertex equal. The rule of the reflection can be expressed as follows:

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

Preimage → Image

A(-1,-4) → A'(-(-1),-4)= A'(1,-4)

B(-3,-2) → B'(-(-3),-2)= B'(3,-2)

C(-1,2) → C'(-(-1),2)= C'(1,2)

After the reflection over the y-axis, the coordinates for the triangle are A'(1,-4), B'(3,-2), and C'(1,2).

ΔABC and ΔA'B'C' are congruent.

Dilation by scale factor k=1/2 ΔA'B'C' to ΔA''B''C''

To dilate a figure by a determined scale factor, you have to multiply the coordinates of each vertex by the said scale factor, you can write the dilation rule as follows:

Dilation factor k=1/2

`(x,y)\to((1)/(2)x,(1)/(2)y)`
`\begin{gathered} \text{Preimage \rightarrow Image} \\ A^(\prime)(1,-4)\to A^(\doubleprime)((1)/(2)\cdot1,(1)/(2)\cdot(-4))=A^(\doubleprime)((1)/(2),-2) \\ B^(\prime)(3,-2)\to B^(\doubleprime)((1)/(2)\cdot3,(1)/(2)(-2))=B^(\doubleprime)((3)/(2),-1) \\ C^(\prime)(1,2)\to C^(\doubleprime)((1)/(2)\cdot1,(1)/(2)\cdot2)=C^(\doubleprime)((1)/(2),1) \end{gathered}`

After the dilation, the coordinates for the new triangle are A''(1/2,-2), B''(3/2,-1), and C''(1/2,1).

ΔABC and ΔA''B''C'' are similar.