Question

Triangle ABC is drawn with vertices at A(−1, −4), B(−6, −6), C(−4, −2). After a rotation, image A′B′C′ has vertices A′(−4, 1), B′(−6, 6), C′(−2, 4).

Part A: Determine the two different rotations that would create the image. (1 point)

Part B: Explain how you know your answer to Part A is correct. (3 points)

Answer 1

Oh, let's embark on a geometric journey with rotations!

Part A: To determine the two different rotations that would create the image, we need to analyze the transformation from the original triangle ABC to the image triangle A'B'C'.

Rotation Option 1: A counterclockwise rotation of 90 degrees about the point (-3, -1) would create the image triangle. This rotation would take each vertex of the original triangle to its corresponding vertex in the image triangle.

Rotation Option 2: A clockwise rotation of 270 degrees about the point (-3, -1) would also result in the image triangle. This rotation would have the same effect as the counterclockwise rotation of 90 degrees.

Part B: I can confidently say that the two rotations mentioned above are correct based on the following reasons:

1. The coordinates of each vertex in the image triangle A'B'C' can be obtained by applying the rotation formulas correctly. The coordinates of each vertex in the original triangle ABC are rotated by the specified angles around the given center point (-3, -1) to obtain the corresponding vertices in the image triangle.

2. The distance between the center of rotation (-3, -1) and each vertex of the original and image triangles remains the same. This is a property of rotations, where the distance between points and the center of rotation is preserved.

3. The angles between the sides of the original triangle ABC are maintained in the image triangle A'B'C'. The relative orientation of the sides and angles of the triangle is preserved under rotations.

By considering these properties and applying the rotation formulas correctly, we can be confident that the mentioned rotations are indeed the ones that would create the image triangle.

Enjoy exploring the fascinating world of geometric transformations, my dear! ✨

Answer 2

Two potential rotations that map triangle ABC to A'B'C' are a 90-degree counterclockwise rotation or a 270-degree clockwise rotation about the origin.

Step-by-step explanation:

To solve this problem, we look for rotations that map the original triangle ABC to the new triangle A'B'C'. For Part A, we can observe that the rotation should maintain the distances between the points and only change their orientation. When we compare the given points, it appears that the center of rotation is at the origin (0,0) since the points are at the same radial distances from the origin before and after the rotation. The specific rotations that map to the new coordinates could be a 90-degree rotation (counterclockwise) or a 270-degree rotation (clockwise).

For Part B, we can confirm these rotations if we apply them to the original points and get the resulting points. A 90-degree counterclockwise rotation would transform point A(-1, -4) to A'(-4, 1), point B(-6, -6) to B'(-6, 6), and point C(-4, -2) to C'(-2, 4). Similarly, a 270-degree clockwise rotation would yield the same results, confirming that these two different rotations can result in the image A'B'C'.