Ask a New Question
Menu

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)

3 answers

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

make it short

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