A line passes through the origin and makes an acute angle theta with the positive x-axis such that tantheta =...

1. TRANSLATE the problem information

• Given information:
  • Line passes through origin
  • Makes acute angle \(\theta\) with positive x-axis
  • \(\tan \theta = \sqrt{3}\)
  • Point P on this line has x-coordinate 125
  • Find distance from origin to P

2. INFER the key relationship

• Since \(\tan \theta\) equals the slope of a line making angle \(\theta\) with the x-axis
• The line's slope = \(\tan \theta = \sqrt{3}\)
• Since the line passes through origin: \(\mathrm{y = (slope)x = \sqrt{3}x}\)

3. SIMPLIFY to find P's coordinates

• Point P has x-coordinate 125
• Substitute into line equation: \(\mathrm{y = \sqrt{3} \times 125 = 125\sqrt{3}}\)
• So \(\mathrm{P = (125, 125\sqrt{3})}\)

4. INFER the distance approach and SIMPLIFY the calculation

• Distance from origin = \(\sqrt{x^2 + y^2}\)
• Distance = \(\sqrt{125^2 + (125\sqrt{3})^2}\)
• Distance = \(\sqrt{125^2 + 125^2 \times 3}\)
  = \(\sqrt{125^2(1 + 3)}\)
  = \(\sqrt{125^2 \times 4}\)
• Distance = \(\sqrt{4 \times 125^2}\)
  = \(\mathrm{2 \times 125 = 250}\)

Answer: D) 250

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect that \(\tan \theta\) represents the slope of the line.

They may know that \(\tan \theta = \sqrt{3}\), but fail to realize this means the line has slope \(\sqrt{3}\). Instead, they might try to use \(\tan \theta\) directly in the distance calculation or get confused about how to find the y-coordinate of point P. This leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly find that \(\mathrm{P = (125, 125\sqrt{3})}\) but make calculation errors when finding the distance.

They might calculate \(\sqrt{125^2 + (125\sqrt{3})^2}\) incorrectly, perhaps forgetting to square the \(\mathrm{125\sqrt{3}}\) term properly or not recognizing that \(\mathrm{125^2(1 + 3) = 125^2 \times 4}\). This could lead them to select Choice C (\(\mathrm{125\sqrt{3}}\)) if they use \(\mathrm{125\sqrt{3}}\) as the distance instead of properly applying the distance formula.

The Bottom Line:

This problem requires connecting trigonometry (\(\tan \theta\) as slope) with coordinate geometry (line equations and distance formula). Students often struggle with the conceptual bridge between these topics.