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a = 5, -1

b = 5, -9
c = 2, -7

Rotate △ABC
 90 degrees (counterclockwise) about the origin to create △A′B′C′
. What are the coordinates of the following points?

5 answers

To rotate a point \((x, y)\) counterclockwise by 90 degrees around the origin, you can use the following transformation:

\[ (x, y) \rightarrow (-y, x) \]

Let's apply this transformation to each of the points \(A\), \(B\), and \(C\):

  1. Point A (5, -1):

    • New coordinates: \((-(-1), 5) = (1, 5)\) \[ A' = (1, 5) \]

  2. Point B (5, -9):

    • New coordinates: \((-(-9), 5) = (9, 5)\) \[ B' = (9, 5) \]

  3. Point C (2, -7):

    • New coordinates: \((-(-7), 2) = (7, 2)\) \[ C' = (7, 2) \]

Thus, the coordinates of points \(A'\), \(B'\), and \(C'\) after the 90-degree counterclockwise rotation about the origin are:

  • \(A' (1, 5)\)
  • \(B' (9, 5)\)
  • \(C' (7, 2)\)
Now reflect  △A′B′C′
 across the y-axis to create  △A"B"C"
. What are the new coordinates

To reflect a point \((x, y)\) across the y-axis, you change the sign of the x-coordinate. The transformation is given by:

\[ (x, y) \rightarrow (-x, y) \]

Let's apply this reflection to each of the points \(A'\), \(B'\), and \(C'\):

  1. Point A' (1, 5):

    • New coordinates: \((-1, 5)\) \[ A" = (-1, 5) \]

  2. Point B' (9, 5):

    • New coordinates: \((-9, 5)\) \[ B" = (-9, 5) \]

  3. Point C' (7, 2):

    • New coordinates: \((-7, 2)\) \[ C" = (-7, 2) \]

Thus, the coordinates of points \(A"\), \(B"\), and \(C"\) after reflecting across the y-axis are:

  • \(A" (-1, 5)\)
  • \(B" (-9, 5)\)
  • \(C" (-7, 2)\)
true or false ABC is congruent to A"B"C"

To determine whether triangles \(ABC\) and \(A"B"C"\) are congruent, we need to consider the transformations that were applied to triangle \(ABC\) to obtain triangle \(A"B"C"\).

  1. Transformations applied:

    • \(△ABC\) was first rotated 90 degrees counterclockwise about the origin to obtain \(△A'B'C'\).
    • Then, \(△A'B'C'\) was reflected across the y-axis to obtain \(△A"B"C"\).

  2. Congruence and Transformations:

    • Both rotations and reflections are rigid transformations, meaning they preserve the size and shape of the geometric figures. Hence, if triangle \(ABC\) undergoes a sequence of rotations and reflections, the resulting triangle will be congruent to the original triangle.

Therefore, since triangles \(ABC\) and \(A"B"C"\) have gone through rigid transformations (rotation and reflection), we can conclude:

True: \(ABC\) is congruent to \(A"B"C"\).