Triangle ABC is equilateral with side length s. The altitude from A to side BC divides triangle ABC into two...

1. TRANSLATE the problem information

• Given information:
  • Equilateral triangle ABC with side length \(\mathrm{s}\)
  • Altitude from A to BC creates two congruent right triangles
  • Perimeter of one right triangle = \(21 + 7\sqrt{3}\)
• What we need to find: The value of \(\mathrm{s}\)

2. INFER what type of triangles are created

• When an altitude is drawn from a vertex to the opposite side in an equilateral triangle, it creates two congruent right triangles
• Since the original triangle has \(60°\) angles and the altitude creates a \(90°\) angle, these are \(30-60-90\) triangles
• In \(30-60-90\) triangles, the sides are in the ratio \(1 : \sqrt{3} : 2\)

3. INFER the side lengths of each right triangle

• If the equilateral triangle has side length \(\mathrm{s}\), then each right triangle has:
  • Short leg (opposite \(30°\)): \(\mathrm{s}/2\) (half the base)
  • Long leg (opposite \(60°\)): \(\mathrm{s}\sqrt{3}/2\) (the altitude)
  • Hypotenuse (opposite \(90°\)): \(\mathrm{s}\) (original side of equilateral triangle)

4. TRANSLATE the perimeter condition into an equation

• Perimeter = \(\mathrm{s}/2 + \mathrm{s}\sqrt{3}/2 + \mathrm{s}\)
• Factor out \(\mathrm{s}\): \(\mathrm{s}(1/2 + \sqrt{3}/2 + 1) = \mathrm{s}(3/2 + \sqrt{3}/2) = \mathrm{s}(3 + \sqrt{3})/2\)
• Set equal to given perimeter: \(\mathrm{s}(3 + \sqrt{3})/2 = 21 + 7\sqrt{3}\)

5. SIMPLIFY to solve for s

• Multiply both sides by 2: \(\mathrm{s}(3 + \sqrt{3}) = 42 + 14\sqrt{3}\)
• Factor the right side: \(\mathrm{s}(3 + \sqrt{3}) = 14(3 + \sqrt{3})\)
• Divide both sides by \((3 + \sqrt{3})\): \(\mathrm{s} = 14\)

Answer: 14

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the \(30-60-90\) triangle relationship

Students may know that an equilateral triangle has equal sides and \(60°\) angles, but fail to connect that the altitude creates \(30-60-90\) triangles with specific side ratios. Without this insight, they might try to use the Pythagorean theorem with unknown variables, leading to a much more complex system of equations. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Algebraic manipulation errors when factoring

Students correctly set up \(\mathrm{s}(3 + \sqrt{3})/2 = 21 + 7\sqrt{3}\) but struggle with factoring \(42 + 14\sqrt{3}\). They might not recognize this as \(14(3 + \sqrt{3})\), leading them to attempt division with mixed radicals and integers. This creates messy fractions and typically results in abandoning the systematic solution and guessing.

The Bottom Line:

This problem requires both geometric insight (recognizing special right triangles) and algebraic skill (factoring expressions with radicals). The key breakthrough is realizing that the altitude in an equilateral triangle always creates \(30-60-90\) triangles with predictable side ratios.