Question

triangle ghj is graphed on a coordinate plane. the transformations listed below are performed on triangle ghj.

• r: reflection across the line x = - 4
• t: translation so that (x, y)→(x + 1, y - 2)

select the letter that corresponds to the triangle shown on the coordinate plane below that is the result of performing transformation r, followed by transformation t, on triangle ghj.
a triangle a
b triangle b
c triangle c
d triangle d

Explanation:

Step1: Perform reflection

For a point $(x,y)$ reflected across the line $x = - 4$, the formula for the $x$-coordinate of the new - point is $x'=-4-(x + 4)=-8 - x$, and the $y$-coordinate remains the same, $y' = y$.

Step2: Perform translation

After the reflection, we perform the translation $(x,y)\to(x + 1,y - 2)$. So the final transformation for a point $(x,y)$ is $x''=-8 - x+1=-7 - x$ and $y''=y - 2$.
We can also analyze the transformation by looking at the key - points of the triangle. Let's assume a vertex of triangle $GHJ$ has coordinates $(x,y)$. After reflection across $x=-4$, its $x$ - coordinate changes as described above, and then after translation, we add 1 to the $x$ - coordinate and subtract 2 from the $y$ - coordinate.
By applying these transformations to the vertices of triangle $GHJ$ and comparing with the given triangles $A$, $B$, $C$, and $D$ on the coordinate - plane, we find the correct triangle.

Answer:

(Without specific coordinate values of the vertices of $\triangle GHJ$, we can't give a definite letter - answer. But the general process is as above. If we assume we have the coordinates of the vertices of $\triangle GHJ$ and calculate one by one: Let the vertices of $\triangle GHJ$ be $(x_1,y_1),(x_2,y_2),(x_3,y_3)$. After reflection across $x = - 4$, the vertices become $(-8 - x_1,y_1),(-8 - x_2,y_2),(-8 - x_3,y_3)$. After translation, they become $(-7 - x_1,y_1 - 2),(-7 - x_2,y_2 - 2),(-7 - x_3,y_3 - 2)$. Then we match with the given triangles on the plane to get the answer.)