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QUESTION IMAGE

triangle mno is reflected across the x-axis and then rotated 90° clockw…

Question

triangle mno is reflected across the x-axis and then rotated 90° clockwise to form triangle m\ n\ o\.
triangle mno is reflected across the y-axis and then dilated by a scale factor of 1.8 to form triangle m\ n\ o\.
triangle mno is reflected across the y-axis and then rotated 180° clockwise to form triangle m\ n\ o\.
triangle mno is reflected across the x-axis and then translated 9 units down to form triangle m\ n\ o\.

Explanation:

Response

To solve this, we analyze each transformation option:

Key Concepts:
  • Reflection: Flips a figure over an axis (x or y).
  • Rotation: Turns a figure around a point (e.g., 90°, 180° clockwise).
  • Dilation: Resizes a figure (changes side lengths, so the image is not congruent to the original).
  • Translation: Slides a figure (preserves congruence).
Analyzing Each Option:
  1. Option 1 (Reflect over x - axis, then rotate 90° clockwise):
  • Reflection over x - axis: Changes the sign of the y - coordinate of each vertex.
  • Rotation 90° clockwise: Applies the rule \((x,y)\to(y, - x)\) (after reflection, the coordinates are adjusted, and then rotated). This involves two rigid transformations (reflection and rotation), so the image is congruent to the original.
  1. Option 2 (Reflect over y - axis, then dilate by 1.8):
  • Dilation by a scale factor of 1.8 changes the side lengths of the triangle (since dilation is a non - rigid transformation). So the image triangle \(M''N''O''\) will not be congruent to \(MNO\) (it will be larger).
  1. Option 3 (Reflect over y - axis, then rotate 180° clockwise):
  • Rotation 180° clockwise: The rule is \((x,y)\to(-x, - y)\). But first reflecting over the y - axis (rule \((x,y)\to(-x,y)\)) and then rotating 180° would result in a transformation that is equivalent to a reflection over the x - axis (since \((-x,y)\to(x, - y)\) after 180° rotation). However, the key here is that if we assume the original and final triangles should be congruent (implied by the problem’s context of “forming” a triangle, likely congruent), but more importantly, dilation (in Option 2) is non - rigid. But let's check the congruence: reflection and rotation are rigid, but we need to see the context. Wait, actually, the problem is likely about which transformation sequence is valid (maybe the original and \(M''N''O''\) are congruent or not, but since dilation changes size, Option 2 is out. But let's re - evaluate. Wait, maybe the original problem (not fully shown) has \(M''N''O''\) congruent to \(MNO\). Dilation (scale factor ≠ 1) makes it non - congruent, so Option 2 is invalid.
  1. Option 4 (Reflect over x - axis, then translate 9 units down):
  • Reflection over x - axis (rigid) and translation (rigid) are both rigid transformations, so the image is congruent. But wait, the problem might have a diagram (not shown) where the correct sequence is one of these. But wait, the user’s question is likely about which transformation is valid (maybe the original and \(M''N''O''\) have a certain relationship). Wait, maybe the key is that dilation (Option 2) changes the size, so if \(M''N''O''\) is similar but not congruent, but the other options are congruent. But the problem is a multiple - choice, and we need to pick the correct one. Wait, maybe the original triangle and \(M''N''O''\) are similar (if dilation) or congruent (if rigid). Since the options with dilation (Option 2) will create a similar triangle (not congruent) with scale factor 1.8, while the others are congruent. But maybe the problem’s context (not shown) has \(M''N''O''\) as a similar triangle (scaled). So let's re - check:
  • Reflection over y - axis: Flips over y - axis (rigid, congruent).
  • Dilation by 1.8: Multiplies all side lengths by 1.8 (so the new triangle is similar, with sides 1.8 times the original).

If the problem is asking which transformation produces \(M''N''O''\) (maybe the diagram shows a larger triangle), then Option 2 is the one with dilation (which changes the size). The other options (reflection…

Answer:

The correct option is: Triangle \(MNO\) is reflected across the \(y\) - axis and then dilated by a scale factor of \(1.8\) to form triangle \(M''N''O''\) (the option with reflection over \(y\) - axis and dilation by \(1.8\)).