Question

which type of transformation maps triangle gfh onto triangle gfh?
○ translation
× reflection
○ rotation
○ dilation

Explanation:

Step1: Recall Transformation Definitions

• Translation: Shifting without rotation/reflection, same orientation.
• Reflection: Mirror image over a line.
• Rotation: Turning around a point.
• Dilation: Resizing (scaling).

Step2: Analyze Triangle Positions

Compare triangle \( GFH \) (red) and \( G'F'H' \) (gray). The orientation (angle of sides, vertex order) suggests a turn (rotation) rather than reflection (mirror) or translation (slide) or dilation (size change, but size looks same here). Wait, initial marked as reflection, but maybe error? Wait, no—wait, looking at the triangles: the red triangle \( GFH \) and gray \( G'F'H' \). Let's check the correspondence: \( G \to G' \), \( F \to F' \), \( H \to H' \). The shape is rotated (like turned around a point) rather than reflected (which would flip over a line, changing left-right or top-bottom). Wait, maybe the initial mark was wrong? Wait, no—wait, the options: translation (slide, same direction), reflection (flip), rotation (turn), dilation (resize). The triangles have same size (so dilation no), and the orientation is changed as if rotated, not reflected. Wait, maybe the user is asking to correct? Wait, the question is "Which type...", and the options. Let's re-express:

Wait, the red triangle \( GFH \): vertices \( F \) (left), \( G \) (top), \( H \) (bottom-right). The gray triangle \( G'F'H' \): \( F' \) (left), \( G' \) (top-right), \( H' \) (bottom). So the triangle is rotated (like 90 degrees or so) around a point. So the correct transformation is rotation, not reflection. Wait, maybe the initial mark (red X on reflection) was incorrect. So the correct answer should be rotation? Wait, no—wait, maybe I missee. Let's check again:

Original triangle: \( F \) (bottom-left), \( G \) (top), \( H \) (bottom-right). The transformed triangle: \( F' \) (top-left), \( G' \) (top-right), \( H' \) (bottom). Wait, maybe it's a rotation. Alternatively, maybe translation? No, translation would keep the same orientation. Reflection would flip over a line, making a mirror image. Rotation would turn. Dilation changes size. Since the size is same, dilation is out. The orientation is changed (the triangle is turned), so rotation. Wait, but the initial marked reflection is wrong. So the correct answer is rotation? Wait, no—wait, maybe the triangles are congruent (same size), so it's a rigid transformation (translation, reflection, rotation). Let's see the vector: if we translate, the direction would be same. If we reflect, there's a line of symmetry. If we rotate, there's a center. Looking at the positions: \( F \) to \( F' \): not a slide (translation), not a flip (reflection), but a turn (rotation). So the correct answer is rotation. Wait, but the options: translation, reflection, rotation, dilation. So the correct option is rotation. Wait, but the initial marked reflection is incorrect. So the correct answer is rotation (option C: rotation).

Wait, maybe the user made a mistake in marking, but the question is to find the correct transformation. So let's go step by step:

1. Check size: same, so dilation (no).
2. Check orientation: the triangle is rotated (turned) relative to original, not reflected (flipped) or translated (slid). So rotation is the correct transformation.

So the correct answer is rotation (option: rotation, which is the third option: "rotation").

Answer:

C. rotation