In right triangle ABC, the right angle is at vertex C, and the sine of acute angle A is 0.6....

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC: right triangle with right angle at C, \(\mathrm{sin(A) = 0.6}\)
  • Triangle XYZ: right triangle with right angle at Z, \(\mathrm{sin(X) = \frac{3}{5}}\)
• Question asks: What additional info needed to determine if triangles are similar?

2. TRANSLATE the sine values to compare them

• Convert 0.6 to fraction form: \(\mathrm{0.6 = \frac{6}{10} = \frac{3}{5}}\)
• Now we can see: \(\mathrm{sin(A) = \frac{3}{5}}\) and \(\mathrm{sin(X) = \frac{3}{5}}\)
• The sine values are equal!

3. INFER what equal sine values mean for the angles

• Since A and X are both acute angles in right triangles, and their sine values are identical, the angles themselves must be equal
• Therefore: \(\mathrm{\angle A \cong \angle X}\)

4. INFER the similarity relationship using angle information

• We now know:
  • \(\mathrm{\angle C \cong \angle Z}\) (both are \(\mathrm{90°}\) right angles)
  • \(\mathrm{\angle A \cong \angle X}\) (from equal sine values)
• This gives us two pairs of congruent corresponding angles

5. INFER that we have enough information for similarity

• By the AA similarity postulate, if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar
• We have satisfied this condition with our two pairs of congruent angles

Answer: D (No additional information is needed)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize that the sine values are equal but fail to connect this to angle congruence, or they don't realize that angle information alone can establish similarity.

Many students think: "We have angle information, but don't we need to know actual side lengths to prove triangles are similar?" This misconception stems from confusion about different similarity criteria - they may only remember SAS or SSS similarity and forget about AA similarity.

This may lead them to select Choice A, B, or C (thinking they need side length information).

The Bottom Line:

This problem tests whether students understand that trigonometric ratios provide angle information, and that angle relationships alone can establish triangle similarity through the AA criterion. The key insight is recognizing that equal sine values for acute angles means the angles are congruent.