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Triangle ABC is rotated -90degrees ( clockwise) about the origin to A'(7,4), B'(-3,5), and C'(1,-2). Find the coordinates of triangle ABC.

A(4,-7), B(5,3), and C(2,1)
A(-4,-7), B(-5,3), and C(2,-1)
A(-7,-4), B(3,-5), and C (-1,2)
A(-4,7), B(-5,-3), and C(2,1)

3 answers

To find the coordinates of triangle ABC after it has been rotated -90 degrees (clockwise) about the origin to yield points A'(7, 4), B'(-3, 5), and C'(1, -2), we need to reverse the rotation.

The transformation for a point \((x, y)\) rotated -90 degrees (clockwise) about the origin can be expressed as: \[ (x', y') = (y, -x) \] where \((x', y')\) are the new coordinates after the rotation.

To find the original coordinates \((x, y)\), we can reverse this transformation: \[ (x, y) = (-y', x') \]

Now, let's apply this transformation to each of the points A', B', and C':

  1. For point A'(7, 4): \[ A: (x, y) = (-y', x') = (-4, 7) \]

  2. For point B'(-3, 5): \[ B: (x, y) = (-y', x') = (-5, -3) \]

  3. For point C'(1, -2): \[ C: (x, y) = (-y', x') = (2, 1) \]

Thus, the coordinates of triangle ABC are:

  • A(-4, 7)
  • B(-5, -3)
  • C(2, 1)

This matches with the option: A(-4, 7), B(-5, -3), and C(2, 1).

Are you sure?

Let’s go through the problem step by step to ensure we have the correct transformation and coordinates for triangle ABC.

To reconstruct the original points A, B, and C from the transformed points A', B', and C' after a -90 degrees (clockwise) rotation, we need to apply the correct inverse transformation.

When we rotate a point \((x, y)\) -90 degrees (clockwise), the transformation is given by: \[ (x', y') = (y, -x) \] To find the coordinates \((x, y)\) of the original triangle ABC corresponding to the new coordinates A'(7, 4), B'(-3, 5), and C'(1, -2), we need to reverse this transformation.

That means: \[ (x, y) = (-y', x') \]

Now, let's apply this to each of the transformed points:

  1. For A'(7, 4): \[ A: (x, y) = (-y', x') = (-4, 7) \]

  2. For B'(-3, 5): \[ B: (x, y) = (-y', x') = (-5, -3) \]

  3. For C'(1, -2): \[ C: (x, y) = (-y', x') = (2, 1) \]

Now let's summarize the original coordinates we derived:

  • A(-4, 7)
  • B(-5, -3)
  • C(2, 1)

Here’s the list of options you provided:

  1. A(4,-7), B(5,3), and C(2,1)
  2. A(-4,-7), B(-5,3), and C(2,-1)
  3. A(-7,-4), B(3,-5), and C (-1,2)
  4. A(-4,7), B(-5,-3), and C(2,1)

The correct coordinates derived from the calculations are indeed:

  • A(-4, 7)
  • B(-5, -3)
  • C(2, 1)

Based on this, the valid option from your list is: A(-4, 7), B(-5, -3), and C(2, 1).

Thank you for your patience, and I confirm that this is the correct solution!