The area of a triangle is equal to x^2 square centimeters. The length of the base of the triangle is...

1. TRANSLATE the problem information

• Given information:
  • Area of triangle = \(\mathrm{x^2}\) square centimeters
  • Base of triangle = \(\mathrm{(2x + 22)}\) centimeters
  • Height of triangle = \(\mathrm{(x - 10)}\) centimeters
  • Need to find: value of x

2. INFER the approach

• Since we have expressions for area, base, and height, we can use the triangle area formula
• The key insight: Set up an equation using \(\mathrm{A = \frac{1}{2} \times base \times height}\)
• This gives us: \(\mathrm{x^2 = \frac{1}{2} \times (2x + 22) \times (x - 10)}\)

3. SIMPLIFY by expanding the right side

• First, expand \(\mathrm{(2x + 22)(x - 10)}\):
  • Distribute: \(\mathrm{2x(x - 10) + 22(x - 10)}\)
  • \(\mathrm{= 2x^2 - 20x + 22x - 220}\)
  • \(\mathrm{= 2x^2 + 2x - 220}\)
• Now our equation becomes:
  \(\mathrm{x^2 = \frac{1}{2}(2x^2 + 2x - 220)}\)
  \(\mathrm{x^2 = x^2 + x - 110}\)

4. SIMPLIFY to solve for x

• Subtract \(\mathrm{x^2}\) from both sides:
  \(\mathrm{0 = x - 110}\)
• Therefore: \(\mathrm{x = 110}\)

Answer: 110

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making sign errors or distribution mistakes when expanding \(\mathrm{(2x + 22)(x - 10)}\)

Students often get confused with the signs, writing something like:
\(\mathrm{(2x + 22)(x - 10) = 2x^2 - 20x - 22x - 220 = 2x^2 - 42x - 220}\)

This leads to the wrong equation \(\mathrm{x^2 = x^2 - 21x - 110}\), which simplifies to \(\mathrm{21x = -110}\), giving \(\mathrm{x \approx -5.2}\). Since this doesn't make geometric sense (negative height), this leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Forgetting the \(\mathrm{\frac{1}{2}}\) factor in the triangle area formula

Some students write \(\mathrm{x^2 = (2x + 22)(x - 10)}\) directly, missing the crucial \(\mathrm{\frac{1}{2}}\). This leads to:
\(\mathrm{x^2 = 2x^2 + 2x - 220}\)
\(\mathrm{-x^2 + 2x - 220 = 0}\)
\(\mathrm{x^2 - 2x + 220 = 0}\)

Using the quadratic formula gives complex solutions, causing confusion and random guessing.

The Bottom Line:

This problem tests whether students can accurately translate a word problem into an equation and then execute algebraic manipulations without sign errors. The key challenge is maintaining precision through multiple algebraic steps while keeping track of the triangle area formula structure.