An isosceles right triangle has legs of length 13 centimeters. Which expression gives the area, in square centimeters, of the...

1. TRANSLATE the problem information

• Given information:
  • Isosceles right triangle with legs of length 13 cm
  • Need to find the area

• What this tells us: We have a right triangle where both legs are equal (13 cm each)

2. INFER the correct approach

• Since this asks for area of a triangle, I need the triangle area formula
• For a right triangle: \(\mathrm{Area = \frac{1}{2} \times base \times height}\)
• The two legs of a right triangle serve as the base and height
• Since both legs are 13 cm: \(\mathrm{Area = \frac{1}{2} \times 13 \times 13}\)

3. Match to the answer choices

Looking at the options:

1. \(\mathrm{2 \cdot 13}\) - This is 26, which looks like adding the legs
2. \(\mathrm{4 \cdot 13}\) - This is 52, which doesn't match our formula
3. \(\mathrm{\frac{1}{2} \cdot 13 \cdot 13}\) - This exactly matches our area calculation
4. \(\mathrm{13 \cdot 13}\) - This is missing the \(\mathrm{\frac{1}{2}}\) factor needed for triangle area

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about shape: Students see "13 by 13" and think "square"

They might reason: "If it has sides of 13, the area is \(\mathrm{13 \times 13}\)." This completely ignores that it's a triangle, not a square. Triangle area always needs the \(\mathrm{\frac{1}{2}}\) factor.

This leads them to select Choice D (\(\mathrm{13 \cdot 13}\))

Second Most Common Error:

Weak INFER skill: Students confuse area with perimeter calculations

They might think: "Two legs of 13 each, so \(\mathrm{2 \times 13}\)" without recognizing this gives a length measurement, not an area. They haven't connected that area needs to involve multiplying two dimensions and including the triangle factor.

This may lead them to select Choice A (\(\mathrm{2 \cdot 13}\))

The Bottom Line:

The key insight is recognizing this is asking for triangle area (requiring the 1/2 factor), not square area or any perimeter calculation. Students who remember the triangle area formula and correctly identify what serves as base and height will succeed.