To determine the new coordinates of the vertices [tex]\( D, E, \)[/tex] and [tex]\( F \)[/tex] after translating the triangle [tex]\( DEF \)[/tex] 5 units up and 6 units to the right, we follow these steps:
1. Identify the initial coordinates:
- [tex]\( D(3, 5) \)[/tex]
- [tex]\( E(6, -6) \)[/tex]
- [tex]\( F(1, 3) \)[/tex]
2. Apply the translation:
- Move each vertex 6 units to the right. This means we add 6 to the x-coordinate of each vertex.
- Move each vertex 5 units up. This means we add 5 to the y-coordinate of each vertex.
3. Calculate the new coordinates:
- For vertex [tex]\( D(3, 5) \)[/tex]:
- New x-coordinate: [tex]\( 3 + 6 = 9 \)[/tex]
- New y-coordinate: [tex]\( 5 + 5 = 10 \)[/tex]
- New coordinates of [tex]\( D \)[/tex] are [tex]\( D(9, 10) \)[/tex].
- For vertex [tex]\( E(6, -6) \)[/tex]:
- New x-coordinate: [tex]\( 6 + 6 = 12 \)[/tex]
- New y-coordinate: [tex]\( -6 + 5 = -1 \)[/tex]
- New coordinates of [tex]\( E \)[/tex] are [tex]\( E(12, -1) \)[/tex].
- For vertex [tex]\( F(1, 3) \)[/tex]:
- New x-coordinate: [tex]\( 1 + 6 = 7 \)[/tex]
- New y-coordinate: [tex]\( 3 + 5 = 8 \)[/tex]
- New coordinates of [tex]\( F \)[/tex] are [tex]\( F(7, 8) \)[/tex].
4. Summarize the new coordinates:
- [tex]\( D(9, 10) \)[/tex]
- [tex]\( E(12, -1) \)[/tex]
- [tex]\( F(7, 8) \)[/tex]
Therefore, after translating the triangle 5 units up and 6 units to the right, the new coordinates of [tex]\( D, E, \)[/tex] and [tex]\( F \)[/tex] are [tex]\( D(9, 10) \)[/tex], [tex]\( E(12, -1) \)[/tex], and [tex]\( F(7, 8) \)[/tex].
Hence, the correct option is:
[tex]\[ D_c(9,10), E_c(12,-1), F_c(7,8) \][/tex]
