The area of a triangle is 270 square centimeters. The length of the base of the triangle is 12 centimeters...

1. TRANSLATE the problem information

• Given information:
  • Area = 270 square cm
  • Base = height + 12 cm
  • Find: height in cm
• Let \(\mathrm{h = height}\), then \(\mathrm{base = h + 12}\)

2. INFER the solution approach

• We have area and know the relationship between base and height
• Use the triangle area formula to create an equation with one variable (height)
• This will give us a quadratic equation to solve

3. TRANSLATE into mathematical equation

• Area formula: \(\mathrm{A = \frac{1}{2} \times base \times height}\)
• Substitute known values: \(\mathrm{270 = \frac{1}{2}(h + 12)(h)}\)

4. SIMPLIFY to solve for height

• \(\mathrm{270 = \frac{(h + 12)h}{2}}\)
• Multiply both sides by 2: \(\mathrm{540 = (h + 12)h}\)
• Expand: \(\mathrm{540 = h^2 + 12h}\)
• Rearrange: \(\mathrm{h^2 + 12h - 540 = 0}\)

5. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to -540 and add to 12
• Those numbers are 30 and -18
• Factored form: \(\mathrm{(h + 30)(h - 18) = 0}\)
• Solutions: \(\mathrm{h = -30}\) or \(\mathrm{h = 18}\)

6. APPLY CONSTRAINTS to select valid solution

• Since height must be positive in the real world: \(\mathrm{h = 18}\) cm

7. INFER the final answer

• The problem asks for the height, which is 18 cm
• (Note: The base would be \(\mathrm{18 + 12 = 30}\) cm, but that's not what's being asked)

Answer: B. 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor INFER reasoning: Students solve the quadratic correctly but then get confused about what the problem is asking for. They calculate both the height (18 cm) and base (30 cm), but then select the base length instead of the height.

This reasoning error leads them to select Choice C (30) even though they did most of the math correctly.

Second Most Common Error:

Weak SIMPLIFY execution: Students make calculation errors when factoring the quadratic \(\mathrm{h^2 + 12h - 540 = 0}\). They might factor incorrectly or make arithmetic mistakes when finding numbers that multiply to -540.

These calculation errors can lead to incorrect factor pairs, causing them to arrive at wrong height values and select Choice A (15) or Choice D (36).

The Bottom Line:

This problem combines algebraic manipulation with careful reading comprehension. Students must not only set up and solve a quadratic equation accurately, but also pay close attention to exactly what quantity the problem is asking them to find.