Certainly! To find the coordinates of the vertices of the image of triangle [tex]\( QRS \)[/tex] after a translation by the vector [tex]\( T_{-76,43} \)[/tex], we need to perform the following steps for each vertex:
1. Translation Operation: A translation by the vector [tex]\( T_{-76,43} \)[/tex] means that every point [tex]\( (x, y) \)[/tex] in the triangle will be moved by [tex]\( -76 \)[/tex] units in the x-direction and [tex]\( +43 \)[/tex] units in the y-direction.
2. Apply Translation to Each Vertex: Add the translation vector [tex]\( (-76, 43) \)[/tex] to the coordinates of each vertex [tex]\( (x, y) \)[/tex].
Let's go through this for each vertex.
For Vertex [tex]\( Q (8, -6) \)[/tex]:
[tex]\[
Q' = \left( 8 + (-76), \ -6 + 43 \right)
\][/tex]
[tex]\[
Q' = (8 - 76, -6 + 43)
\][/tex]
[tex]\[
Q' = (-68, 37)
\][/tex]
For Vertex [tex]\( R (10, 5) \)[/tex]:
[tex]\[
R' = \left( 10 + (-76), \ 5 + 43 \right)
\][/tex]
[tex]\[
R' = (10 - 76, 5 + 43)
\][/tex]
[tex]\[
R' = (-66, 48)
\][/tex]
For Vertex [tex]\( S (-3, 3) \)[/tex]:
[tex]\[
S' = \left( -3 + (-76), \ 3 + 43 \right)
\][/tex]
[tex]\[
S' = (-3 - 76, 3 + 43)
\][/tex]
[tex]\[
S' = (-79, 46)
\][/tex]
So, the coordinates of the vertices of the image of the triangle [tex]\( QRS \)[/tex] after the translation [tex]\( T_{-76,43}(x, y) \)[/tex] are:
[tex]\[
Q' = (-68, 37)
\][/tex]
[tex]\[
R' = (-66, 48)
\][/tex]
[tex]\[
S' = (-79, 46)
\][/tex]
This gives us the final transformed coordinates for each vertex.
