An isosceles triangle has a base of 336 millimeters. The perimeter of the triangle is 3,856 millimeters. What is the...

1. TRANSLATE the problem information

• Given information:
  • Isosceles triangle with \(\mathrm{base = 336\ mm}\)
  • \(\mathrm{Perimeter = 3,856\ mm}\)
  • Need altitude to the base

2. INFER what we need to find first

• In an isosceles triangle, two sides are equal
• To use the altitude property, we need to know the length of these equal sides
• Strategy: Find equal side lengths, then use the right triangle created by the altitude

3. Find the equal side lengths

• Let each equal side = s
• Perimeter equation: \(\mathrm{s + s + base = total\ perimeter}\)
• \(\mathrm{2s + 336 = 3,856}\)
• \(\mathrm{2s = 3,520}\)
• \(\mathrm{s = 1,760\ mm}\)

4. INFER the right triangle setup

• The altitude to the base in an isosceles triangle bisects the base
• This creates two identical right triangles
• Each right triangle has:
  • Hypotenuse = \(\mathrm{1,760\ mm}\) (one equal side)
  • One leg = \(\mathrm{336 \div 2 = 168\ mm}\) (half the base)
  • Other leg = h (the altitude we want)

5. SIMPLIFY using Pythagorean theorem

• \(\mathrm{h^2 + 168^2 = 1,760^2}\)
• \(\mathrm{h^2 = 1,760^2 - 168^2}\)
• \(\mathrm{h^2 = 3,097,600 - 28,224 = 3,069,376}\)
• \(\mathrm{h = \sqrt{3,069,376} = 1,752\ mm}\) (use calculator)

Answer: B. 1,752

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to use the altitude formula directly without first finding the equal side lengths.

They might attempt to work with just the base and perimeter, not realizing they need the individual side lengths to form the right triangle. This leads to confusion about what values to use in the Pythagorean theorem, causing them to get stuck and guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Students incorrectly set up the right triangle by confusing which measurement is the hypotenuse.

They might use the base (336 mm) as the hypotenuse instead of the equal side (1,760 mm), or use the full base instead of half the base (168 mm) as one leg. This leads to incorrect Pythagorean calculations and may lead them to select Choice C (1,760) or Choice A (168).

The Bottom Line:

This problem requires understanding the structural relationship between an isosceles triangle's parts and how the altitude creates a right triangle with specific measurements. Students must work systematically: sides first, then triangle geometry.