Question

Triangle QRS has coordinates Q(-8,4), R(-5,8) and S(-3,3). Triangle QRS is transformed by (x,y)→(x+4,y−6) to produce △Q′R′S′ . Then △Q′R′S′ is transformed by the rule (x,y)→(x+5,y−3) to produce △Q"R"S"
A. What is the final images will be in what quadrant?
B. Create the rule for the composition transformation. example (x+2,y-4)

Answer 1

To determine the final images of triangle \( QRS \) after the specified transformations, we'll begin by applying the transformations step by step.

### Step 1: Apply the first transformation

The first transformation is given by:

\[
(x,y) \rightarrow (x + 4, y - 6)
\]

We will apply this transformation to the vertices of triangle \( QRS \) with coordinates \( Q(-8,4) \), \( R(-5,8) \), and \( S(-3,3) \).

1. For \( Q(-8, 4) \):
\[
Q' = (-8 + 4, 4 - 6) = (-4, -2)
\]

2. For \( R(-5, 8) \):
\[
R' = (-5 + 4, 8 - 6) = (-1, 2)
\]

3. For \( S(-3, 3) \):
\[
S' = (-3 + 4, 3 - 6) = (1, -3)
\]

After the first transformation, the coordinates of triangle \( Q'R'S' \) are:
- \( Q'(-4, -2) \)
- \( R'(-1, 2) \)
- \( S'(1, -3) \)

### Step 2: Apply the second transformation

The second transformation is given by:

\[
(x,y) \rightarrow (x + 5, y - 3)
\]

Now, we apply this transformation to the coordinates of triangle \( Q'R'S' \).

1. For \( Q'(-4, -2) \):
\[
Q'' = (-4 + 5, -2 - 3) = (1, -5)
\]

2. For \( R'(-1, 2) \):
\[
R'' = (-1 + 5, 2 - 3) = (4, -1)
\]

3. For \( S'(1, -3) \):
\[
S'' = (1 + 5, -3 - 3) = (6, -6)
\]

After the second transformation, the coordinates of triangle \( Q''R''S'' \) are:
- \( Q''(1, -5) \)
- \( R''(4, -1) \)
- \( S''(6, -6) \)

### A. What Quadrant will the final images be in?
The final coordinates are:
- \( Q''(1, -5) \) is in Quadrant IV (positive x, negative y).
- \( R''(4, -1) \) is also in Quadrant IV (positive x, negative y).
- \( S''(6, -6) \) is also in Quadrant IV (positive x, negative y).

All three points \( Q'', R'', S'' \) are in Quadrant IV.

### B. Create the rule for the composition transformation
To find the overall transformation from the original coordinates to the final coordinates, we can compose the two transformations:

1. First transformation: \((x, y) \rightarrow (x + 4, y - 6)\)
2. Second transformation: \((x, y) \rightarrow (x + 5, y - 3)\)

We can find the combined transformation by adding the changes:

- For the \( x \)-coordinate:
\[
x + 4 + 5 = x + 9
\]

- For the \( y \)-coordinate:
\[
y - 6 - 3 = y - 9
\]

Thus, the rule for the composition of the transformations is:

\[
(x, y) \rightarrow (x + 9, y - 9)
\]

### Final Summary
**A.** The final images \( Q'', R'', S'' \) will all be in Quadrant IV.

**B.** The rule for the composition transformation is:
\[
(x, y) \rightarrow (x + 9, y - 9)
\]