Question

1. which is a true statement about the transformation shown?

a. the graph shows a translation, and the figures are congruent.
b. the graph shows a reflection, and the figures are congruent.
c. the graph shows a translation, and the figures are similar.
d. the graph shows a reflection, and the figures are similar.
(there is also some other text and options about dilations, and a coordinate grid image with two triangles labeled o, m, n and o, m, n)

Explanation:

Step1: Identify Transformation Type

A translation is a slide (horizontal/vertical shift) without reflection/rotation/dilation. The triangles here have the same shape, size, and orientation, shifted right and down? Wait, looking at coordinates: M(-6,2), M'(2,1)? Wait no, grid: M is at (-6,2), N at (-2,2); M' at (2,1)? Wait no, the graph: original triangle M(-6,2), N(-2,2), O(-4,7); transformed M'(2,1)? Wait no, maybe I misread. Wait, the x-axis: M is at x=-6, M' at x=2 (shift right 8? No, wait the grid lines: M is at (-6,2), M' at (2,1)? Wait no, maybe the y-coordinate: original O is at (-4,7), O' at (4,7)? Wait, looking at the graph, the triangles are shifted horizontally (translation) because the shape, size, and orientation are the same. Translation preserves congruence (so also similarity, but congruent implies similar).

Step2: Analyze Congruence/Similarity

Translation is a rigid transformation (preserves distance, angle, so congruent). Congruent figures are also similar (since similarity is same shape, congruent is same shape and size). Now check options:

A: Translation, figures congruent. Let's verify: translation is rigid, so congruent. B: Reflection? No, reflection would flip, but here orientation same (triangle direction same). C: Translation, similar (but congruent is stronger, and A says congruent which is true). Wait, wait maybe I made a mistake. Wait the coordinates: M(-6,2), M'(2,1)? No, maybe the y-axis: original M is at (-6,2), M' at (2,1)? Wait no, the grid: each square is 1 unit. Original triangle: base MN from x=-6 to x=-2 (length 4), height from y=2 to y=7 (height 5). Transformed triangle: M' at x=2, N' at x=6 (length 4), height from y=1 to y=6 (height 5). Wait, y-coordinate: original O at (-4,7), O' at (4,6)? No, maybe the y-axis is flipped? Wait no, the triangles look identical, just shifted. So translation (since no reflection, rotation, dilation). Translation preserves congruence (so figures are congruent, hence similar). Now options:

A: Translation, congruent. B: Reflection? No, same orientation. C: Translation, similar (but congruent is more accurate, and A says congruent). Wait but maybe the y-coordinate: original M is at (-6,2), M' at (2,1)? Wait, maybe the graph has a typo, but visually, the triangles are translated (shifted) and congruent. So A is correct? Wait no, wait the answer might be A. Wait but let's recheck:

Translation: rigid motion, so congruent. Congruent figures are similar. So A says translation and congruent, which is true. B: reflection? No, same orientation. C: translation and similar (but A is more precise). D: reflection? No. So the correct answer is A? Wait but maybe I messed up. Wait the original triangle: M(-6,2), N(-2,2), O(-4,7). Transformed: M'(2,1), N'(6,1), O'(4,6). Wait, y-coordinate: original y=2, transformed y=1 (shift down 1), x: shift right 8. So translation (shift right 8, down 1). So it's a translation, and since translation is rigid, figures are congruent. So option A: "The graph shows a translation, and the figures are congruent." is true.

Answer:

A. The graph shows a translation, and the figures are congruent.