To analyze the given transformations on triangle ΔABC with vertices A(−8,4), B(−5,8), and C(−3,3), let's first break down the transformations step by step.
1. **First transformation: (x,y) → (x + 4, y - 6)**
We apply this transformation to each vertex of ΔABC.
- A(−8, 4):
\[ A' = (−8 + 4, 4 - 6) = (−4, -2) \]
- B(−5, 8):
\[ B' = (−5 + 4, 8 - 6) = (−1, 2) \]
- C(−3, 3):
\[ C' = (−3 + 4, 3 - 6) = (1, -3) \]
After the first transformation, the coordinates of ΔA'B'C' are:
- A'(-4, -2)
- B'(-1, 2)
- C'(1, -3)
2. **Second transformation: (x,y) → (x + 5, y - 3)**
Now we apply the second transformation to the new vertices A', B', and C'.
- A'(-4, -2):
\[ A'' = (−4 + 5, −2 - 3) = (1, -5) \]
- B'(-1, 2):
\[ B'' = (−1 + 5, 2 - 3) = (4, -1) \]
- C'(1, -3):
\[ C'' = (1 + 5, −3 - 3) = (6, -6) \]
Now, we have the coordinates of ΔA"B"C":
- A''(1, -5)
- B''(4, -1)
- C''(6, -6)
3. **Composition of transformations:**
The overall transformation can be represented as follows:
- Original point (x, y) → (x + 4, y - 6) → (x + 4 + 5, y - 6 - 3)
- Combining these we get:
\[ (x, y) → (x + 4 + 5, y - 6 - 3) \]
\[ (x, y) → (x + 9, y - 9) \]
Thus, the rule for the composition transformation is:
\[ (x, y) → (x + 9, y - 9) \]
4. **Prediction of the quadrant:**
Now let's check the coordinates of ΔA"B"C":
- A''(1, -5)
- B''(4, -1)
- C''(6, -6)
All points have a positive x-coordinate and a negative y-coordinate. This means the points are located in the fourth quadrant.
Thus, the final coordinates of ΔA"B"C will be A(1, -5), B(4, -1), C(6, -6), and it will be located in the **fourth quadrant**.