Question:A line L passes through the origin and the point 12, 5.The coordinate plane is dilated by a factor of...

1. TRANSLATE the problem information

• Given information:
  • Line L passes through origin and \(12,5\)
  • Plane dilated by factor 7 → creates line L'
  • Need to find \(\sin(\theta')\) where \(\theta'\) is angle L' makes with positive x-axis

2. INFER the starting approach

• Since we need \(\sin(\theta')\), we first need to understand the angle \(\theta'\) that L' makes
• Key insight: Let's first find the angle \(\theta\) that original line L makes, then see how dilation affects it

3. TRANSLATE slope to angle relationship

• Slope of L = \(\frac{5-0}{12-0} = \frac{5}{12}\)
• For any line through the origin: slope = \(\tan(\theta)\)
• Therefore: \(\tan(\theta) = \frac{5}{12}\)

4. VISUALIZE using right triangle

• Draw right triangle with:
  • Opposite side = 5
  • Adjacent side = 12
  • Hypotenuse = \(\sqrt{5^2 + 12^2} = \sqrt{169} = 13\)

5. SIMPLIFY to find sine

• \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}\)

6. INFER the effect of dilation

• Under dilation by factor 7: \(12,5\) becomes \(84,35\)
• Slope of L' = \(\frac{35}{84} = \frac{5}{12}\)
• This is identical to slope of L!

7. INFER the key insight

• Same slope means same angle with x-axis
• Therefore: \(\theta' = \theta\)
• So: \(\sin(\theta') = \sin(\theta) = \frac{5}{13}\)

Answer: \(\frac{5}{13}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that dilation preserves slope and therefore angle. They might think the dilation changes the angle, leading them to try complex calculations involving the new coordinates \(84,35\) to find a "different" angle. This leads to confusion and potentially incorrect calculations, causing them to guess or select an incorrect fractional answer.

Second Most Common Error:

Poor TRANSLATE reasoning: Students struggle to connect slope with trigonometric ratios. They might find the slope correctly but then don't know how to use \(\tan(\theta) = \frac{5}{12}\) to find \(\sin(\theta)\). Without this connection, they get stuck trying to work directly with coordinates rather than using the right triangle approach.

The Bottom Line:

The key insight that makes this problem elegant is recognizing that dilation about the origin preserves angles. Students who miss this end up doing much more complex work than necessary and often make calculation errors in the process.