A square-based pyramid has a square base with side length 12 units. The height of the pyramid from the apex...

1. VISUALIZE the pyramid structure

• Given information:
  • Square base with side length 12 units
  • Height from apex to base center: 5 units
  • Need: slant height from apex to midpoint of any base edge

• This problem requires seeing the right triangle hidden within the 3D pyramid

2. INFER the key measurement needed

• To find slant height, I need the distance from base center to the midpoint of any edge
• Since the base is square with side length 12, this distance is 6 units (half the side length)
• This creates a right triangle with the height and slant height

3. INFER the triangle relationship

• Right triangle components:
  • Vertical leg (height): 5 units
  • Horizontal leg (center to edge midpoint): 6 units
  • Hypotenuse (slant height): unknown

4. SIMPLIFY using Pythagorean theorem

• \(\mathrm{slant\ height}^2 = 5^2 + 6^2\)
• \(\mathrm{slant\ height}^2 = 25 + 36 = 61\)
• \(\mathrm{slant\ height} = \sqrt{61}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak VISUALIZE skill: Students struggle to identify the right triangle within the 3D pyramid structure. They may try to use the full side length (12) instead of the distance from center to edge midpoint (6), leading to:

\(\mathrm{slant\ height}^2 = 5^2 + 12^2 = 25 + 144 = 169\)

\(\mathrm{slant\ height} = \sqrt{169} = 13\)

This may lead them to select Choice D (13)

Second Most Common Error:

Poor INFER reasoning: Students may recognize they need a right triangle but incorrectly identify which measurements to use, perhaps trying to use the height and side length directly without understanding the geometric relationships.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem requires strong 3D visualization skills to see the "hidden" right triangle formed by connecting the apex to the base center and then to an edge midpoint. Without this spatial understanding, students cannot set up the correct Pythagorean relationship.