Sure, let's determine the new coordinates of the vertices of triangle QRS after applying the translation defined by [tex]\( T_{-7.6, 4.3}(x, y) \)[/tex].
1. Defining the Translation Vector:
The translation vector is [tex]\( (-7.6, 4.3) \)[/tex], which means each vertex of the triangle will be shifted left by 7.6 units and up by 4.3 units.
2. Translating Vertex [tex]\( Q \)[/tex]:
- Original coordinates of [tex]\( Q \)[/tex] are [tex]\( (8, -6) \)[/tex].
- Apply the translation:
[tex]\[
Q' = \left(8 + (-7.6), -6 + 4.3\right)
\][/tex]
- Calculate the new x-coordinate:
[tex]\[
8 - 7.6 = 0.4
\][/tex]
- Calculate the new y-coordinate:
[tex]\[
-6 + 4.3 = -1.7
\][/tex]
- Therefore, the new coordinates of [tex]\( Q' \)[/tex] are:
[tex]\[
Q' = (0.4, -1.7)
\][/tex]
3. Translating Vertex [tex]\( R \)[/tex]:
- Original coordinates of [tex]\( R \)[/tex] are [tex]\( (10, 5) \)[/tex].
- Apply the translation:
[tex]\[
R' = \left(10 + (-7.6), 5 + 4.3\right)
\][/tex]
- Calculate the new x-coordinate:
[tex]\[
10 - 7.6 = 2.4
\][/tex]
- Calculate the new y-coordinate:
[tex]\[
5 + 4.3 = 9.3
\][/tex]
- Therefore, the new coordinates of [tex]\( R' \)[/tex] are:
[tex]\[
R' = (2.4, 9.3)
\][/tex]
4. Translating Vertex [tex]\( S \)[/tex]:
- Original coordinates of [tex]\( S \)[/tex] are [tex]\( (-3, 3) \)[/tex].
- Apply the translation:
[tex]\[
S' = \left(-3 + (-7.6), 3 + 4.3\right)
\][/tex]
- Calculate the new x-coordinate:
[tex]\[
-3 - 7.6 = -10.6
\][/tex]
- Calculate the new y-coordinate:
[tex]\[
3 + 4.3 = 7.3
\][/tex]
- Therefore, the new coordinates of [tex]\( S' \)[/tex] are:
[tex]\[
S' = (-10.6, 7.3)
\][/tex]
Thus, the coordinates of the vertices after the translation are:
- [tex]\( Q' = (0.4, -1.7) \)[/tex]
- [tex]\( R' = (2.4, 9.3) \)[/tex]
- [tex]\( S' = (-10.6, 7.3) \)[/tex]
So the final transformed vertices of the triangle after the translation [tex]\( T_{-7.6, 4.3} \)[/tex] are:
[tex]\[
Q ^{\prime}=\checkmark(0.4, -1.7), R ^{\prime}=\checkmark(2.4, 9.3), S ^{\prime}=\checkmark(-10.6, 7.3).
\][/tex]
