An isosceles triangle has two equal sides of length sqrt(65) and a base of length 8. What is the area...

1. TRANSLATE the problem information

• Given information:
  • Isosceles triangle with two equal sides of length \(\sqrt{65}\)
  • Base of length 8
  • Need to find the area

2. INFER the strategic approach

• To find area, I need: \(\mathrm{Area} = \frac{1}{2} \times \mathrm{base} \times \mathrm{height}\)
• I have the base (8), but need to find the height
• Key insight: In an isosceles triangle, the altitude from the vertex angle to the base bisects the base and creates two right triangles

3. VISUALIZE the right triangle setup

• Draw the altitude from the top vertex to the base
• This creates two identical right triangles
• Each right triangle has:
  • Hypotenuse = \(\sqrt{65}\) (one of the equal sides)
  • One leg = 4 (since altitude bisects the 8-unit base)
  • Other leg = h (the height we need)

4. SIMPLIFY using Pythagorean theorem

• Apply \(\mathrm{a}^2 + \mathrm{b}^2 = \mathrm{c}^2\):

\(\mathrm{h}^2 + 4^2 = (\sqrt{65})^2\)

\(\mathrm{h}^2 + 16 = 65\)

\(\mathrm{h}^2 = 49\)

\(\mathrm{h} = 7\)

5. SIMPLIFY the area calculation

• \(\mathrm{Area} = \frac{1}{2} \times \mathrm{base} \times \mathrm{height}\)
• \(\mathrm{Area} = \frac{1}{2} \times 8 \times 7 = 28\)

Answer: (B) 28

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to find the height first, or don't see that drawing an altitude creates useful right triangles.

Without this key insight, students might attempt to use the triangle area formula directly with the given side lengths, which doesn't work. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the Pythagorean theorem correctly but make arithmetic errors, such as:

• Calculating \(\sqrt{49} = 6\) instead of 7
• Making errors in the final area calculation

These calculation mistakes could lead them to select Choice (A) (16) or Choice (C) (32) instead of the correct answer.

The Bottom Line:

This problem requires students to connect their knowledge of isosceles triangle properties with right triangle relationships. The breakthrough moment is realizing that the altitude creates right triangles that can be solved using the Pythagorean theorem.