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Triangle

is shown on the coordinate plane.


​
​Triangle
is rotated
counterclockwise about the origin to form the image triangle


(not shown). The triangle


is reflected across the
axis to form triangle


(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 points \( A \) and \( B \) without knowing their specific locations, we need to consider the transformations (rotation and reflection) described in the question.

Part A: Signs of the coordinates of point \( A \)

  1. Triangle is rotated counterclockwise about the origin:

    • This transformation does not change the signs of the coordinates relative to the quadrants they occupy, but it will affect the actual coordinates depending on the angle of rotation.
    • However, if point \( A \) starts in quadrant I, it'll stay in quadrant I after rotation; if it starts in quadrant II, it will move to quadrant III, and so on.

  2. Without any specific details of point \( A \)’s initial position before the rotation, we can’t definitively state the signs. However, let's consider potential cases:

    • If point \( A \) is in quadrant I initially, then after transformation, it remains in quadrant I.
    • If point \( A \) starts in quadrants II, III, IV, we would need further details to determine if it remains positive or switches.

Given the possible options:

  • If point \( A \) is in quadrant I, it could result in: (A) Both \( x \) and \( y \) are positive.
  • If it begins in quadrant II, it could lead to: (B) \( x \) is negative and \( y \) is positive.
  • If it starts in quadrant III, it leads to: (C) Both \( x \) and \( y \) are negative.
  • If it starts in quadrant IV, it results in: (D) \( x \) is positive and \( y \) is negative.

Without knowing the origin of point \( A \), we conclude that the answer will depend on its initial quadrant.

Part B: Signs of the coordinates of point \( B \)

  1. Triangle is reflected across the x-axis:
    • This transformation changes the sign of the \( y \)-coordinate while keeping the \( x \)-coordinate unchanged.
    • If point \( B \) was above the x-axis (with \( y > 0 \)), it would move below it (to \( y < 0 \)).
    • Conversely, if it was the opposite, it would flip to above the x-axis.

Similar to Part A, specifics about point B’s location are crucial:

  • If point \( B \) starts with positive \( y \), it will end with negative \( y \): possible outcome could be: (D) \( x \) is positive and \( y \) is negative.
  • If its original position in the respective quadrants and transformations were different, any of the outcomes based on its original and reflective coordinates would hold true.

In summary, without specific coordinates provided for points \( A \) and \( B \), we cannot definitively answer the signs of the coordinates for both parts. The answers can only be determined with more information regarding their initial positions.