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"

Question

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