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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 (x,y)
of point P'?​
A Both x and y are positive.
B x is negative and y is positive
C Both x and y are negative.
D x is positive and y is negative.


Part B
​What are the signs of the coordinates (x,y)
of point Q''?​
A Both x and y are positive.
B x is negative and y is positive.
C
Both x and y are negative.
D x is positive and y is negative.

1 answer

To answer the questions regarding the transformations of triangle PQR, let's go through the steps systematically.

Part A: Determine the signs of the coordinates of point P' after a 90-degree counterclockwise rotation about the origin.

When a point \((x, y)\) is rotated 90 degrees counterclockwise about the origin, the new coordinates \((x', y')\) are given by the transformation: \[ (x', y') = (-y, x) \]

  1. Original coordinates of point P: (let's assume \(P = (x, y)\))

    After the 90-degree counterclockwise rotation: \[ P' = (-y, x) \]

  2. Depending on the signs of \(y\) and \(x\):

    • If \(y > 0\) and \(x > 0\), then \(P' = (-y, x)\) results in \(x\) being positive and \(y\) being negative. Thus:
      • \(P'\) has negative x and positive y, giving the option D.
    • If \(y < 0\) or \(x < 0\) (and similarly analyzed), you can see that one coordinate will be positive even if one is negative.

In general, without specific coordinates for P, it's difficult to ascertain the exact sign. However, in many triangle transformation problems, if \(P\) is in the first quadrant, then after transformation, \(P'\) results in \(x\) being negative while \(y\) is still positive.

Thus: Answer for Part A: B (x is negative and y is positive).

Part B: Determine the signs of the coordinates of point Q'' after reflecting point Q' across the x-axis.

After reflection across the x-axis, the coordinates will transform as follows: \[ (x'', y'') = (x', -y') \]

We start from point \(Q'\) which, after rotating \(Q\) 90 degrees counterclockwise: \[ Q' = (-y_Q, x_Q) \]

On reflection across the x-axis: \[ Q'' = (-y_Q, -x_Q) \]

Now, the signs of the coordinates will depend on the original coordinates of point \(Q\):

  1. If \(x_Q > 0\) and \(y_Q > 0\), then:
    • \(Q''\) becomes negative x and negative y, giving us coordinate signs for Q'' as both negative.
  2. If \(x_Q\) or \(y_Q\) takes other values, we would derive accordingly.

In many cases, if we assume Q to be in the first quadrant:

  • Then \(Q''\) will indeed have both coordinates negative.

Answer for Part B: C (Both x and y are negative).

If you provide specific coordinates for points P and Q, I could confirm this with more precision!