The problem asks which sequence of transformations maps triangle $LMN$ onto triangle $PQR$, with four answer options provided. Let’s analyze the transformations involved.

1. Observing the coordinates of both triangles:

• Triangle $LMN$ is smaller and positioned lower on the graph.
  • $L(0, -6)$, $M(2, -4)$, $N(4, -6)$.
• Triangle $PQR$ is larger and higher up.
  • $P(0, 4)$, $Q(8, 7)$, $R(4, 4)$.

2. Identifying the transformations:

Dilation:

We can first check if dilation is a possible transformation. Comparing the lengths of corresponding sides:

• $LN = MN = 4$ for triangle $LMN$,
• $PR = PQ = 8$ for triangle $PQR$.

The sides of triangle $PQR$ are twice the lengths of those in triangle $LMN$, indicating a dilation by a scale factor of 2.

Translation:

Next, consider a translation. The corresponding point $L(0, -6)$ maps to point $P(0, 4)$. This represents a vertical translation 10 units upwards (from $-6$ to $4$).

3. Conclusion:

A dilation by a scale factor of 2 followed by a translation 10 units up maps triangle $LMN$ onto triangle $PQR$.

However, none of the options exactly match this sequence. The closest option that seems to capture this is "a dilation by a scale factor of 2 about the origin followed by a translation 7 units up," although the translation specified is slightly off.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you calculate the scale factor in transformations?
2. What are the key differences between dilation and translation?
3. How do you determine if two triangles are similar by transformation?
4. How does rotation affect the orientation of a triangle on a graph?
5. How do you calculate the result of a reflection over the x-axis?

Tip: Always check if corresponding points have the same relative positions when performing a dilation or translation.