Question

identifying transformations what are the vertices for the final image after applying the composition t - 2,4 ∘ r o,180 ∘ to △xyz? x is y is z is

Explanation:

Step1: Determine the original vertices

Assume original vertices of $\triangle XYZ$ are $X(1,5)$, $Y(1,3)$, $Z(3,3)$.

Step2: Apply rotation $R_{O,180^{\circ}}$

The rule for a $180^{\circ}$ rotation about the origin $(x,y)\to(-x,-y)$. So $X(1,5)\to X_1(-1,-5)$, $Y(1,3)\to Y_1(-1,-3)$, $Z(3,3)\to Z_1(-3,-3)$.

Step3: Apply translation $T_{- 2,4}$

The rule for translation $T_{-2,4}$ is $(x,y)\to(x - 2,y + 4)$. So $X_1(-1,-5)\to X''(-1-2,-5 + 4)=(-3,-1)$, $Y_1(-1,-3)\to Y''(-1-2,-3 + 4)=(-3,1)$, $Z_1(-3,-3)\to Z''(-3-2,-3 + 4)=(-5,1)$.

Answer:

$X''$ is $(-3,-1)$.
$Y''$ is $(-3,1)$.
$Z''$ is $(-5,1)$.