Question

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)

Answer 1

Are you sure?

Answer 2

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)**.

Answer 3

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!