An isosceles triangle has two equal sides of length sqrt(208) and a base of length 16. An altitude is drawn...

1. TRANSLATE the problem information

• Given information:
  • Isosceles triangle with two equal sides of length \(\sqrt{208}\)
  • Base length = 16
  • Need to find altitude from vertex between equal sides to base

2. INFER the geometric setup

• Key insight: When you draw an altitude from the vertex of an isosceles triangle to the base, it creates two congruent right triangles
• The altitude is perpendicular to the base and bisects it
• This means each right triangle has legs of length 8 (half the base) and h (the altitude), with hypotenuse \(\sqrt{208}\)

3. INFER the solution approach

• Since we have a right triangle, we can use the Pythagorean theorem
• Set up: \(\mathrm{h}^2 + 8^2 = (\sqrt{208})^2\)

4. SIMPLIFY to solve for the altitude

• \(\mathrm{h}^2 + 64 = 208\)
• \(\mathrm{h}^2 = 208 - 64 = 144\)
• \(\mathrm{h} = \sqrt{144} = 12\)

Answer: 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that the altitude creates right triangles, or forgetting that the altitude bisects the base in an isosceles triangle.

Students might try to use the altitude as a side of the original triangle rather than recognizing it creates two right triangles with legs of 8 and h. Without this key insight, they get stuck and either guess randomly or try to apply formulas incorrectly.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors in the calculations, particularly when computing 208 - 64 or finding \(\sqrt{144}\).

Students might incorrectly calculate \(208 - 64 = 154\) (instead of 144) or make errors when taking the square root. These calculation mistakes lead to wrong final answers even when the setup is correct.

The Bottom Line:

This problem tests whether students can visualize how an altitude divides an isosceles triangle into right triangles and then correctly apply the Pythagorean theorem. The key breakthrough is recognizing the geometric relationship, not just memorizing formulas.