Two similar triangular support brackets are manufactured with proportional dimensions. The first bracket has a base of 15 cm and...

1. TRANSLATE the problem information

• Given information:
  • First bracket: base = 15 cm, height = 10 cm
  • Second bracket: base = 9 cm, height = unknown
  • The brackets are similar (key word!)

2. INFER the mathematical relationship

• Similar figures have proportional corresponding sides
• This means: \(\frac{\mathrm{base_1}}{\mathrm{height_1}} = \frac{\mathrm{base_2}}{\mathrm{height_2}}\)
• We can set up a proportion to find the missing height

3. TRANSLATE into mathematical notation

Set up the proportion with corresponding sides:
\(\frac{15}{10} = \frac{9}{\mathrm{h}}\)

Where h is the unknown height of the second bracket.

4. SIMPLIFY by cross multiplying

• Cross multiply: \(15 \times \mathrm{h} = 10 \times 9\)
• This gives us: \(15\mathrm{h} = 90\)
• Divide both sides by 15: \(\mathrm{h} = \frac{90}{15} = 6\)

Answer: B. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing what "similar" means mathematically, or setting up the proportion incorrectly by mismatching corresponding sides.

Students might set up the proportion as \(\frac{15}{9} = \frac{10}{\mathrm{h}}\) (base to base = height to height instead of keeping the ratios of base to height consistent). This incorrect setup leads to:
\(15\mathrm{h} = 90\), so \(\mathrm{h} = 6\)... wait, that actually gives the same answer by coincidence!

Let me reconsider: they might set up \(\frac{15}{9} = \frac{\mathrm{h}}{10}\), giving \(15 \times 10 = 9\mathrm{h}\), so \(150 = 9\mathrm{h}\), and \(\mathrm{h} = \frac{150}{9} \approx 16.67\). Since this isn't among the choices, this leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic mistakes during cross multiplication or division.

For example, incorrectly calculating \(15\mathrm{h} = 90\) as \(\mathrm{h} = \frac{90}{10} = 9\) instead of \(\mathrm{h} = \frac{90}{15} = 6\). Since 9 isn't an answer choice, this causes them to get stuck and guess.

The Bottom Line:

The key insight is recognizing that "similar" creates a proportional relationship between corresponding sides, and then maintaining consistency in how you set up that proportion.