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)

(4 points)

Answer 1

To determine the final image of triangle QRS after the two transformations, we'll follow these steps:

### Step 1: Apply the first transformation

**Original coordinates:**
- Q(-8, 4)
- R(-5, 8)
- S(-3, 3)

**First transformation:** \((x, y) → (x + 4, y - 6)\)

**Calculate new coordinates:**
- \(Q'(-8 + 4, 4 - 6) = Q'(-4, -2)\)
- \(R'(-5 + 4, 8 - 6) = R'(-1, 2)\)
- \(S'(-3 + 4, 3 - 6) = S'(1, -3)\)

### Step 2: Apply the second transformation

**Coordinates after the first transformation:**
- \(Q'(-4, -2)\)
- \(R'(-1, 2)\)
- \(S'(1, -3)\)

**Second transformation:** \((x, y) → (x + 5, y - 3)\)

**Calculate new coordinates:**
- \(Q''(-4 + 5, -2 - 3) = Q''(1, -5)\)
- \(R''(-1 + 5, 2 - 3) = R''(4, -1)\)
- \(S''(1 + 5, -3 - 3) = S''(6, -6)\)

### Part a: Determine the quadrant for triangle \(Q''R''S''\)

**Final coordinates:**
- \(Q''(1, -5)\)
- \(R''(4, -1)\)
- \(S''(6, -6)\)

All final coordinates have positive x-values and negative y-values. Therefore, they lie in the **Fourth Quadrant**.

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

To create the rule for the combined transformations, we need to add the transformations of both steps.

**First transformation:** \((x, y) → (x + 4, y - 6)\)

**Second transformation:** \((x, y) → (x + 5, y - 3)\)

### Combining the transformations:

1. For x-coordinate: \(x + 4 + 5 = x + 9\)
2. For y-coordinate: \(y - 6 - 3 = y - 9\)

The final transformation rule is:

\[
(x, y) → (x + 9, y - 9)
\]

### Summary:

a. The final image, **\(△Q''R''S''\)**, will be in the **Fourth Quadrant**.

b. The combined transformation rule is \((x + 9, y - 9)\).