Question

the triangles are congruent by the sss congruence theorem. which transformation(s) can map △lmn onto △lmn? options: reflection then translation, reflection only, translation only, rotation then translation

Explanation:

Step1: Analyze Triangle Orientation

First, observe the orientation of $\triangle LMN$ and $\triangle L'M'N'$. The original triangle and the image have a rotational change (the triangle is turned) and then a translation (moved) to align. A reflection alone wouldn't account for the rotational aspect, and translation alone can't change the orientation. A reflection then translation or rotation then translation? Wait, let's check the congruence and transformation. The triangles have the same side markings (SSS congruent). To map $\triangle LMN$ to $\triangle L'M'N'$, we can first rotate $\triangle LMN$ to match the orientation of $\triangle L'M'N'$ and then translate it to the correct position. Alternatively, reflection then translation? Wait, no—let's see the vertices. $N$ maps to $N'$, $M$ to $M'$, $L$ to $L'$. The orientation: original triangle has $N$ on the left, $M$ on the right, $L$ top. The image has $N'$ top - left, $M'$ bottom, $L'$ right. So a rotation (like 90 degrees or some angle) to reorient, then translation. So the transformation is rotation then translation. Wait, but let's check the options. The options are: reflection then translation, translation only, rotation then translation, reflection only. Wait, no—wait the options are: "reflection then translation", "translation only", "rotation then translation", "reflection only". Wait, the correct transformation: first, rotate $\triangle LMN$ to get the same orientation as $\triangle L'M'N'$, then translate. So "rotation then translation" is the correct option? Wait, no—wait the triangles: let's see the side markings. $LN$ has two marks, $MN$ three, $LM$ one. In the image, $L'N'$ has two marks, $M'N'$ three, $L'M'$ one. So the correspondence is $L \to L'$, $M \to M'$, $N \to N'$. So to map $L$ to $L'$, $M$ to $M'$, $N$ to $N'$: first, rotate the triangle so that the sides align in orientation, then translate. So the transformation is rotation then translation. So the correct option is "rotation then translation". Wait, but let's check the options again. The options are:

1. reflection then translation

1. translation only

1. rotation then translation

1. reflection only

So the correct one is "rotation then translation".

Step2: Eliminate Other Options

• "translation only": Can't change orientation, so wrong.

• "reflection only": Would flip the triangle, but the orientation here is rotated, not flipped, so wrong.

• "reflection then translation": Reflection would flip, but the image doesn't look like a reflection (the side markings' order: in original, $LN$ (two marks) is left, $MN$ (three) bottom, $LM$ (one) right. In image, $L'N'$ (two) is left - side, $M'N'$ (three) is left - bottom, $L'M'$ (one) is right - side. Wait, maybe a rotation. So "rotation then translation" is correct.

Answer:

rotation then translation (the option with "rotation then translation" text)