Question

find a translation that has the same effect as the composition of translations below.
$t_{(5,5)}(x,y)$ followed by $t_{(-3,9)}(x,y)$
choose the correct answer below.
a. $(x,y)\to(x + 8,y + 14)$
b. $(x,y)\to(x + 8,y - 4)$
c. $(x,y)\to(x + 2,y - 4)$
d. $(x,y)\to(x + 2,y + 14)$

Explanation:

Step1: Analyze the first translation

The translation $T_{(5,5)}(x,y)$ moves a point $(x,y)$ to $(x + 5,y+5)$.

Step2: Analyze the second translation

The translation $T_{(-3,9)}(x,y)$ moves a point $(x,y)$ to $(x-3,y + 9)$. When we apply it after the first - translation, we substitute the result of the first translation into the second. So, if after the first translation we have the point $(x_1,y_1)=(x + 5,y+5)$, then after the second translation, the new point $(x_2,y_2)$ is given by:
$x_2=(x + 5)-3=x+2$
$y_2=(y + 5)+9=y + 14$

Answer:

D. $(x,y)\to(x + 2,y+14)$