Triangles PQR and LMN are graphed in the xy-plane. Triangle PQR has vertices P, Q, and R at \(\mathrm{(4,5)}\), \(\mathrm{(4,7)}\),...

1. TRANSLATE the coordinate information into geometric relationships

• Given Triangle PQR vertices: \(\mathrm{P(4,5)}\), \(\mathrm{Q(4,7)}\), \(\mathrm{R(6,5)}\)
• Given Triangle LMN vertices: \(\mathrm{L(4,5)}\), \(\mathrm{M(4,7+k)}\), \(\mathrm{N(6+k,5)}\)

Let's find the side lengths and orientations:

• Side PQ: from (4,5) to (4,7) → vertical line, length = 2
• Side PR: from (4,5) to (6,5) → horizontal line, length = 2
• Side LM: from (4,5) to (4,7+k) → vertical line, length = 2+k
• Side LN: from (4,5) to (6+k,5) → horizontal line, length = 2+k

2. INFER the triangle properties from these relationships

• Triangle PQR: \(\mathrm{PQ \perp PR}\) (perpendicular) and \(\mathrm{PQ = PR = 2}\)
• This makes PQR a right isosceles triangle with right angle at P

• Triangle LMN: \(\mathrm{LM \perp LN}\) (perpendicular) and \(\mathrm{LM = LN = 2+k}\)
• This makes LMN a right isosceles triangle with right angle at L

3. APPLY CONSTRAINTS of triangle angle relationships

• In right isosceles triangle PQR:
  • \(\mathrm{\angle P = 90°}\) (right angle)
  • \(\mathrm{\angle Q = \angle R = 45°}\) (equal base angles)
• Since \(\mathrm{\angle Q = t°}\), we know \(\mathrm{t = 45°}\)

• In right isosceles triangle LMN:
  • \(\mathrm{\angle L = 90°}\) (right angle)
  • \(\mathrm{\angle M = \angle N = 45°}\) (equal base angles)
• Therefore \(\mathrm{\angle N = 45°}\)

4. INFER the relationship between t and \(\mathrm{\angle N}\)

• We found: \(\mathrm{\angle N = 45°}\) and \(\mathrm{t = 45°}\)
• So \(\mathrm{\angle N = 90° - t°}\)

Answer: C. \(\mathrm{(90-t)°}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that both triangles are right isosceles triangles from their coordinate structure.

Students might calculate individual side lengths correctly but miss the crucial insight that perpendicular sides of equal length create a right isosceles triangle. They may try to use more complex angle formulas or get confused about how the parameter k affects the triangle's shape. Without this geometric insight, they cannot determine that both \(\mathrm{\angle Q}\) and \(\mathrm{\angle N}\) equal 45°.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about coordinate geometry: Misinterpreting how the parameter k transforms triangle LMN.

Students might think that k changes the triangle's fundamental shape or angle relationships, rather than recognizing that k simply scales both perpendicular sides equally. They may attempt complex calculations involving k instead of seeing that the right isosceles property is preserved regardless of k's value.

This may lead them to select Choice D. \(\mathrm{(90+k)°}\) by incorrectly incorporating k into the angle measure.

The Bottom Line:

This problem tests whether students can recognize geometric patterns from coordinate representations and understand that scaling preserves angle relationships in similar triangles.