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Triangle QRS has coordinates Q(-8,4), R(-5,8) and s(-3,3). Triangle QRS is transformed by (x,y) to (x+4, y-6) to produce triangle Q'R'S'. Then triangle Q'R'S' is transformed by the rule (x,y) to (x+5, y-3) to produce Q"R"S"

The final triangle Q"R"S" will be in which quadrant?
Creat the rule for composition transformation. example (x+2,y-4)

1 answer

To determine the final coordinates of triangle \( Q"R"S" \) after the series of transformations, we will apply each transformation step by step to the original triangle \( QRS \) with given vertices \( Q(-8,4) \), \( R(-5,8) \), and \( S(-3,3) \).

Step 1: First Transformation (x,y) to (x+4, y-6)

  1. Transform point \( Q(-8, 4) \): \[ Q' = (-8+4, 4-6) = (-4, -2) \]

  2. Transform point \( R(-5, 8) \): \[ R' = (-5+4, 8-6) = (-1, 2) \]

  3. Transform point \( S(-3, 3) \): \[ S' = (-3+4, 3-6) = (1, -3) \]

After the first transformation, the coordinates of triangle \( Q'R'S' \) are:

  • \( Q'(-4, -2) \)
  • \( R'(-1, 2) \)
  • \( S'(1, -3) \)

Step 2: Second Transformation (x,y) to (x+5, y-3)

  1. Transform point \( Q'(-4, -2) \): \[ Q'' = (-4 + 5, -2 - 3) = (1, -5) \]

  2. Transform point \( R'(-1, 2) \): \[ R'' = (-1 + 5, 2 - 3) = (4, -1) \]

  3. Transform point \( S'(1, -3) \): \[ S'' = (1 + 5, -3 - 3) = (6, -6) \]

After the second transformation, the coordinates of triangle \( Q"R"S" \) are:

  • \( Q''(1, -5) \)
  • \( R''(4, -1) \)
  • \( S''(6, -6) \)

Determining the Quadrant

Now we analyze the final coordinates:

  • \( Q''(1, -5) \) is in the fourth quadrant (x > 0, y < 0).
  • \( R''(4, -1) \) is also in the fourth quadrant (x > 0, y < 0).
  • \( S''(6, -6) \) is in the fourth quadrant (x > 0, y < 0).

Since all points of triangle \( Q"R"S" \) lie in the fourth quadrant, the final triangle will be in the fourth quadrant.

Rule for Composition Transformation

To find the overall transformation rule from the original triangle \( QRS \) to the final triangle \( Q"R"S" \):

  1. First transformation: \( (x, y) \to (x + 4, y - 6) \)
  2. Second transformation: \( (x, y) \to (x + 5, y - 3) \)

To combine these transformations, add the changes for each corresponding component:

  • For the x-coordinate: \( (x + 4) + 5 = x + 9 \)
  • For the y-coordinate: \( (y - 6) - 3 = y - 9 \)

Therefore, the overall rule for the composition of transformations is: \[ (x, y) \to (x + 9, y - 9) \]