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)

Question

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 \((x + 9, y - 9)\).