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. The final image, △Q"R"S"
, will be in which quadrant?

b. Create the rule for the composition transformation. example (x+2,y-4)

Answer 1

To analyze the transformations and where the final image \( \triangle Q''R''S'' \) will be located, we will first apply the transformations step by step.

**Step 1: Apply the first transformation to triangle QRS:**
- The coordinates of triangle \( QRS \) are:
- \( Q(-8, 4) \)
- \( R(-5, 8) \)
- \( S(-3, 3) \)

Applying the transformation \( (x,y) \rightarrow (x+4, y-6) \):

- New coordinates for \( Q' \):
\[
Q' = (-8 + 4, 4 - 6) = (-4, -2)
\]
- New coordinates for \( R' \):
\[
R' = (-5 + 4, 8 - 6) = (-1, 2)
\]
- New coordinates for \( S' \):
\[
S' = (-3 + 4, 3 - 6) = (1, -3)
\]

Now, the new coordinates for \( \triangle Q'R'S' \) are \( Q'(-4, -2), R'(-1, 2), S'(1, -3) \).

**Step 2: Apply the second transformation to triangle \( Q'R'S' \):**
Applying the transformation \( (x,y) \rightarrow (x+5, y-3) \):

- New coordinates for \( Q'' \):
\[
Q'' = (-4 + 5, -2 - 3) = (1, -5)
\]
- New coordinates for \( R'' \):
\[
R'' = (-1 + 5, 2 - 3) = (4, -1)
\]
- New coordinates for \( S'' \):
\[
S'' = (1 + 5, -3 - 3) = (6, -6)
\]

The new coordinates for \( \triangle Q''R''S'' \) are \( Q''(1, -5), R''(4, -1), S''(6, -6) \).

**a. Determining the quadrant of \( \triangle Q''R''S'' \):**

The coordinates \( (1, -5) \), \( (4, -1) \), and \( (6, -6) \):
- All these points have a positive x-coordinate and a negative y-coordinate.
- Therefore, all points lie in the **fourth quadrant**.

**b. Finding the rule for the composition transformation:**

To find the overall transformation from \( (x, y) \) to \( (x', y') \), we need to combine the two transformations step-by-step:

1. **First transformation:**
\[
\begin{align*}
x' &= x + 4 \\
y' &= y - 6
\end{align*}
\]

2. **Second transformation using the result of the first:**
\[
\begin{align*}
x'' &= (x + 4) + 5 = x + 9 \\
y'' &= (y - 6) - 3 = y - 9
\end{align*}
\]

The overall transformation from \( (x, y) \) to \( (x'', y'') \) is:
\[
(x, y) \rightarrow (x + 9, y - 9)
\]

Thus, the rule for the composition transformation is \( (x + 9, y - 9) \).