Right triangle R is similar to right triangle S. The area of triangle R is 162 square centimeters, and the...

1. TRANSLATE the problem information

• Given information:
  • Triangle R and S are similar right triangles
  • Area of R = 162 square cm, Area of S = 18 square cm
  • Hypotenuse of R = 27 cm
• Find: Hypotenuse of S

2. INFER the scaling relationship approach

• Since triangles are similar, their corresponding sides are proportional
• Key insight: Areas of similar figures scale by the square of the linear scale factor
• Strategy: Find linear scale factor from area ratio, then apply to hypotenuse

3. SIMPLIFY to find the linear scale factor

• If linear scale factor from R to S is k, then: \(\mathrm{Area_{S} = k^2 \times Area_{R}}\)
• Substitute: \(\mathrm{18 = k^2 \times 162}\)
• Solve for \(\mathrm{k^2}\): \(\mathrm{k^2 = \frac{18}{162} = \frac{1}{9}}\)
• Take square root: \(\mathrm{k = \sqrt{\frac{1}{9}} = \frac{1}{3}}\)

4. SIMPLIFY to find the hypotenuse of triangle S

• Hypotenuse of S = k × Hypotenuse of R
• Hypotenuse of S = \(\mathrm{\frac{1}{3} \times 27 = 9\text{ cm}}\)

Answer: B (9)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students use the area ratio directly as the linear scale factor instead of recognizing they need to take the square root.

They calculate: Hypotenuse of S = \(\mathrm{\frac{18}{162} \times 27 = \frac{1}{9} \times 27 = 3\text{ cm}}\)

This leads them to select Choice A (3).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make arithmetic errors when calculating the scale factor or applying it.

Some might incorrectly calculate \(\mathrm{\sqrt{\frac{18}{162}}}\) or make errors in the final multiplication, leading to confusion and guessing among the remaining choices.

The Bottom Line:

This problem requires understanding that linear measurements and area measurements scale differently for similar figures - the key insight that trips up many students is remembering to take the square root of the area ratio to get the linear scale factor.