Question

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 solve the problem systematically, let's break it down step-by-step.

### Part a: Determine the final image's quadrant

1. **Coordinates of triangle QRS:**
- \(Q(-8, 4)\)
- \(R(-5, 8)\)
- \(S(-3, 3)\)

2. **First Transformation \((x,y) \to (x+4, y-6)\):**
- For Q:
\[
Q' = (-8+4, 4-6) = (-4, -2)
\]
- For R:
\[
R' = (-5+4, 8-6) = (-1, 2)
\]
- For S:
\[
S' = (-3+4, 3-6) = (1, -3)
\]

So, the coordinates after the first transformation are:
- \(Q'(-4, -2)\)
- \(R'(-1, 2)\)
- \(S'(1, -3)\)

3. **Second Transformation \((x,y) \to (x+5, y-3)\):**
- For Q':
\[
Q'' = (-4+5, -2-3) = (1, -5)
\]
- For R':
\[
R'' = (-1+5, 2-3) = (4, -1)
\]
- For S':
\[
S'' = (1+5, -3-3) = (6, -6)
\]

So, the coordinates after the second transformation are:
- \(Q''(1, -5)\)
- \(R''(4, -1)\)
- \(S''(6, -6)\)

4. **Determine the quadrant:**
- All points have positive x-values and negative y-values:
- \(Q''(1, -5)\) lies in Quadrant IV
- \(R''(4, -1)\) lies in Quadrant IV
- \(S''(6, -6)\) lies in Quadrant IV

**Answer for Part a:** The final image, triangle \(ΔQ''R''S''\), will be located in **Quadrant IV**.

### Part b: Create the rule for the composition transformation

1. **First transformation:**
- \((x, y) \to (x + 4, y - 6)\)

2. **Second transformation:**
- \((x, y) \to (x + 5, y - 3)\)

3. **Composition of transformations:**
- When you apply the first transformation, you transform coordinates \( (x, y) \) into \((x + 4, y - 6)\). Then, apply the second transformation to this output:
\[
(x + 4 + 5, (y - 6) - 3) = (x + 9, y - 9)
\]

Thus, the rule for the composition transformation is:
\[
(x, y) \to (x + 9, y - 9)
\]

**Answer for Part b:** The rule for the composition transformation is \((x + 9, y - 9)\).

Answer 2

make it short

Answer 3

Sure! Here’s a concise version:

### Part a: Quadrant of Triangle \(ΔQ''R''S''\)
After the transformations:
- \(Q''(1, -5)\)
- \(R''(4, -1)\)
- \(S''(6, -6)\)

All points have positive x-values and negative y-values, which means they are in **Quadrant IV**.

### Part b: Rule for Composition Transformation
1. First transformation: \( (x, y) \to (x + 4, y - 6) \)
2. Second transformation: \( (x, y) \to (x + 5, y - 3) \)

Composition:
\[
(x + 4 + 5, y - 6 - 3) = (x + 9, y - 9)
\]

**Final Rule:** \((x + 9, y - 9)\)