To describe a sequence of transformations that maps triangle ABC onto triangle A''B''C'', a student starts with a reflection over the x-axis. How should the student complete the sequence of transformations to map triangle ABC onto triangle A''B''C''?
The student should then rotate the triangle 180 degrees around the origin, followed by a reflection over the y-axis.
To complete the sequence of transformations that maps triangle ABC onto triangle A''B''C'', the student needs to perform two more transformations after the reflection over the x-axis.
1. Start with the reflection over the x-axis. This means that each point's y-coordinate will get negated, while the x-coordinate remains the same.
2. The second transformation is a translation. The student needs to move the triangle horizontally or vertically to align it with the desired position. This can be done by adding or subtracting a specific value from the x and/or y coordinates of every point in the triangle. The specific values should be determined by the desired location of triangle A''B''C''.
3. The last transformation is rotation or a combination of rotation and reflection. To rotate the triangle, the student needs to choose a point (center of rotation) and the desired angle of rotation. The rotation can be clockwise or counterclockwise. If a reflection is also desired, the rotation point needs to lie on the line of reflection.
These three transformations, performed in the given order, will map triangle ABC onto triangle A''B''C''.
To map triangle ABC onto triangle A''B''C'' after reflecting over the x-axis, the student needs to perform additional transformations. Here is a step-by-step sequence of transformations:
1. Reflection over the x-axis: This step has already been completed by the student. The reflection over the x-axis will flip the original triangle ABC so that it lies below the x-axis. Let's call the image of triangle ABC after reflection as A'B'C'.
2. Translation: Now, the student needs to perform a translation to move triangle A'B'C' to align with triangle A''B''C''. This translation will shift the triangle horizontally or vertically. The student should select a point that is common to both triangles (e.g., a vertex or the centroid) and determine the vector between the corresponding points in both triangles. This vector will represent the amount and direction that the student needs to move triangle A'B'C'. Let's say the common point is A', and the vector is v. Now, the student should apply the translation on A'B'C' using the vector v to obtain A''B''C''.
3. Rotation: If the two triangles are not congruent after the translation, the student needs to apply a rotation. To determine the amount of rotation needed, the student should select another common point between the two triangles (e.g., another vertex or the centroid) and determine the angle between the corresponding segments in both triangles. This angle will represent the amount and direction of rotation. Let's say the common point is A'', and the angle is θ. Now, the student should apply the rotation on A''B''C'' using the angle θ to obtain the final mapped triangle A''B''C''.
By following these steps of reflection, translation, and rotation, the student will successfully map triangle ABC onto triangle A''B''C''.