A circle has center G, and points M and N lie on the circle. Line segments MH and NH are...

1. TRANSLATE the problem information

• Given information:
  • Circle with center G, \(\mathrm{radius = 168\text{ mm}}\)
  • Points M and N on the circle
  • MH and NH are tangents to the circle
  • Perimeter of quadrilateral GMHN = 3,856 mm
  • Need to find distance GH

2. INFER key geometric relationships

• Since M and N are on the circle: \(\mathrm{GM = GN = 168\text{ mm}}\) (both radii)
• Since MH and NH are tangents from the same external point H: \(\mathrm{MH = NH}\) (equal tangent property)
• Let x represent the length of each tangent segment

3. SIMPLIFY using the perimeter condition

• \(\mathrm{Perimeter = GM + MH + NH + GN}\)
• \(\mathrm{168 + x + x + 168 = 3,856}\)
• \(\mathrm{336 + 2x = 3,856}\)
• \(\mathrm{2x = 3,520}\)
• \(\mathrm{x = 1,760\text{ mm}}\)
• So \(\mathrm{MH = NH = 1,760\text{ mm}}\)

4. INFER the right triangle setup

• Since MH is tangent at M, it's perpendicular to radius GM
• This creates right triangle GMH with:
  • \(\mathrm{GM = 168\text{ mm}}\) (one leg)
  • \(\mathrm{MH = 1,760\text{ mm}}\) (other leg)
  • GH = hypotenuse (what we want)

5. SIMPLIFY using Pythagorean theorem

• \(\mathrm{GM^2 + MH^2 = GH^2}\)
• \(\mathrm{168^2 + 1,760^2 = GH^2}\)
• \(\mathrm{28,224 + 3,097,600 = GH^2}\) (use calculator for \(\mathrm{1,760^2}\))
• \(\mathrm{3,125,824 = GH^2}\)
• \(\mathrm{GH = \sqrt{3,125,824} = 1,768\text{ mm}}\) (use calculator)

Answer: D. 1,768

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize that GM and GN are radii but fail to connect that tangent segments from an external point have equal length. Without this insight, they can't set up the perimeter equation properly. This leads to confusion and guessing.

Second Most Common Error:

Incomplete solution process: Students correctly find that the tangent segments are 1,760 mm each but then incorrectly assume this is the answer to the question. They don't recognize that they need to use the Pythagorean theorem to find the distance GH. This may lead them to select Choice C (1,760).

The Bottom Line:

This problem requires connecting multiple geometric properties in sequence - equal radii, equal tangents, perpendicular relationships, and the Pythagorean theorem. Students who rush or don't systematically work through each relationship often get derailed partway through.