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Triangle

is shown on the coordinate plane.

image-4317380-1524369096184
​
​Triangle PQR
is rotated 90 degrees counterclockwise about the origin to form the image triangle P'Q'R' is reflected across the x
axis to form triangle P''Q''R''


(not shown).
​
​Part A
​What are the signs of the coordinates
of point
?​
A
Both xxx and yyy are positive.
B
xxx is negative and yyy is positive
C
Both xxx and yyy are negative.
D
xxx is positive and yyy is negative.
(b)

Part B
​What are the signs of the coordinates
of point
?​
A
Both xxx and yyy are positive.
B
xxx is negative and yyy is positive.
C
Both xxx and yyy are negative.
D
xxx is positive and yyy is negative.

1 answer

To determine the signs of the coordinates of the points in triangle \( PQR \) and their transformations, we need to understand how the transformations affect the signs of the coordinates.

  1. Rotating 90 degrees counterclockwise about the origin:

    • A point \((x, y)\) becomes \((-y, x)\) after this transformation.
    • If \(x > 0\) and \(y > 0\), then after the transformation, \((-y, x)\) becomes negative in the x-coordinate and positive in the y-coordinate.
    • If \(x < 0\) and \(y > 0\), then after the transformation, it would keep the y positive but make x negative.
    • If \(x < 0\) and \(y < 0\), both coordinates will remain negative after rotation.

  2. Reflecting across the x-axis:

    • Any point \((x, y)\) becomes \((x, -y)\) after this transformation.
    • This means that the x-coordinate's sign remains the same, and the y-coordinate's sign changes.

Now, let's analyze this for points \(P\) and \(Q\) in parts A and B:

Part A (Point P)

  • Before transformations: Assume the possible coordinates for point \(P\).
  • After rotation: The new coordinates transformed will be \((-y, x)\).
  • After reflection across the x-axis: The new coordinates will be \((-y, -x)\).
  • Depending on the original signs of \(x\) and \(y\), we can deduce the final signs.

Assuming \(P\) initially has positive coordinates, after transformation:

  • If original \( x \) and \( y \) are both positive, \(P'' = (-y, -x)\) would result in both coordinates being negative.
  • If \( P \) has \(x\) positive and \(y\) negative, we can derive a similar conclusion.

Conclusion for Part A:

  • The signs of the coordinates for point \(P\) must be C: Both \(x\) and \(y\) are negative.

Part B (Point Q)

  • Using similar logic for point \(Q\):
  • After rotation, it transforms to \((-y', x')\) and after reflection it becomes \((-y', -x')\).

Conclusion for Part B:

  • Under similar transformations, if both original signs for coordinate \(Q\) are assumed positive, then:
  • The final coordinates would reflect both \(x'\) and \(y'\) negative.

Therefore, for part A the answer is C, and for part B the answer is also C. If you have specific coordinates for triangle \(PQR\), we can directly assign the exact signs to the points to validate this reasoning!