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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?
Create the rule for composition transformation. example (x+2,y-4)

1 answer

To find the final coordinates of triangle Q"R"S" after the two transformations, we can start with the initial coordinates of triangle QRS and apply each transformation one at a time.

Step 1: Transform triangle QRS to triangle Q'R'S' using the rule (x, y) → (x + 4, y - 6)

  • For point Q(-8, 4):

    • Q' = (-8 + 4, 4 - 6) = (-4, -2)

  • For point R(-5, 8):

    • R' = (-5 + 4, 8 - 6) = (-1, 2)

  • For point S(-3, 3):

    • S' = (-3 + 4, 3 - 6) = (1, -3)

So, the coordinates for triangle Q'R'S' are:

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

Step 2: Transform triangle Q'R'S' to triangle Q"R"S" using the rule (x, y) → (x + 5, y - 3)

  • For point Q'(-4, -2):

    • Q" = (-4 + 5, -2 - 3) = (1, -5)

  • For point R'(-1, 2):

    • R" = (-1 + 5, 2 - 3) = (4, -1)

  • For point S'(1, -3):

    • S" = (1 + 5, -3 - 3) = (6, -6)

So, the coordinates for triangle Q"R"S" are:

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

Step 3: Determine the quadrant of triangle Q"R"S"

To determine the quadrant, we can look at the y-coordinates of the points:

  • Q"(1, -5) is in the fourth quadrant (positive x, negative y)
  • R"(4, -1) is in the fourth quadrant (positive x, negative y)
  • S"(6, -6) is also in the fourth quadrant (positive x, negative y)

Since all points of triangle Q"R"S" have positive x-coordinates and negative y-coordinates, triangle Q"R"S" is located entirely in the fourth quadrant.

Step 4: Create the rule for composition transformation

To find the overall transformation rule, we can combine the two transformations:

  1. From QRS to Q'R'S': (x, y) → (x + 4, y - 6)
  2. From Q'R'S' to Q"R"S": (x, y) → (x + 5, y - 3)

We combine the two transformations by adding the increments to x and y:

  • Total transformation for x: \( x + 4 + 5 = x + 9 \)
  • Total transformation for y: \( y - 6 - 3 = y - 9 \)

Thus, the overall transformation rule from triangle QRS to triangle Q"R"S" is: \[ (x, y) \rightarrow (x + 9, y - 9) \]