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