Two triangles, GHI and JKL, are similar. The correspondence is such that angle G corresponds to angle J, and angle...

1. TRANSLATE the problem information

• Given information:
  • Triangles GHI and JKL are similar
  • \(\angle G\) corresponds to \(\angle J\), \(\angle I\) corresponds to \(\angle L\)
  • \(\angle I\) and \(\angle L\) are right angles (90°)
  • \(\mathrm{sec(G)} = 2.6\)
  • Need to find: \(\mathrm{csc(K)}\)

2. INFER the complete angle correspondence

• Since triangles are similar, ALL corresponding angles must be congruent
• We know \(\angle G \cong \angle J\) and \(\angle I \cong \angle L\)
• Therefore, the third pair must be: \(\angle H \cong \angle K\)

3. INFER the angle relationships in triangle GHI

• Since \(\angle I = 90°\), triangle GHI is a right triangle
• In any triangle: \(\angle G + \angle H + \angle I = 180°\)
• Substituting: \(\angle G + \angle H + 90° = 180°\)
• Therefore: \(\angle G + \angle H = 90°\) (complementary angles)

4. INFER the trigonometric connection

• For complementary angles, we use co-function identities
• Since \(\angle G + \angle H = 90°\), we have: \(\mathrm{sec(G)} = \mathrm{csc(H)}\)
• Since \(\angle H \cong \angle K\) from similarity: \(\mathrm{csc(H)} = \mathrm{csc(K)}\)
• Therefore: \(\mathrm{csc(K)} = \mathrm{sec(G)} = 2.6\)

Answer: C) 2.60

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to establish the complete angle correspondence in similar triangles. They might focus only on the given correspondences (\(\angle G \cong \angle J\), \(\angle I \cong \angle L\)) and not realize that \(\angle H \cong \angle K\). Without this connection, they cannot link \(\mathrm{csc(K)}\) to the known information about triangle GHI, leading to confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge about co-function identities: Students might correctly identify that \(\angle G\) and \(\angle H\) are complementary but not remember or apply the relationship \(\mathrm{sec(G)} = \mathrm{csc(H)}\). They may try to calculate trigonometric values directly or use incorrect relationships. This may lead them to select Choice A (0.38) if they mistakenly calculate \(1/\mathrm{sec(G)} = 1/2.6 \approx 0.38\).

The Bottom Line:

This problem tests the intersection of geometric similarity and trigonometric identities. Success requires recognizing that similar triangles preserve all angle relationships, then applying co-function identities to bridge between different trigonometric functions of complementary angles.