Let's begin by analyzing the given problem.

The vertices of the triangles are:

• Triangle $JKL$: $J(1, 1)$, $K(3, 2)$, $L(4, 1)$
• Triangle $MNP$: $M(6, 1)$, $N(5, 2)$, $P(2, 1)$

We need to determine whether these triangles are congruent and, if so, by what transformation.

Step 1: Analyzing Coordinates

We first check whether there is a transformation (translation, rotation, reflection, or rigid motion) that maps triangle $JKL$ onto triangle $MNP$.

• To compare, we start by noticing the structure of the coordinates. We can calculate the horizontal and vertical shifts necessary to match the triangles.

Step 2: Identifying a Transformation

To map triangle $JKL$ to triangle $MNP$:

• Translation: Check if a translation maps the triangles.

  • The horizontal difference between point $J(1, 1)$ and point $M(6, 1)$ is 5 units to the right (from 1 to 6). To move left by 5 units would align point $M(6, 1)$ with point $J(1, 1)$.
  • Similarly, moving the entire triangle $MNP$ 4 units to the left would align point $J(1, 1)$ with $P(2, 1)$.

• Reflection/Rotation: Since the relative orientation of the points doesn't involve rotation or reflection, we can infer that no such transformation is needed.

Conclusion:

The transformation that maps triangle $JKL$ onto triangle $MNP$ is a translation 4 units left. Therefore, the triangles are congruent.

Would you like further details or have any questions?

Follow-up Questions:

1. What does it mean for two triangles to be congruent?
2. How would a reflection affect the orientation of triangle $JKL$?
3. Could a combination of transformations (rotation + translation) work for these triangles?
4. What would happen if the vertices of one triangle were rotated by 90°?
5. How do you calculate the magnitude of a translation between two points?

Tip:

Always double-check the orientation and distances between points when applying transformations like translation or rotation!