In triangle RST, angle T is a right angle, point L lies on RS, point K lies on ST, and...

1. TRANSLATE the problem information

• Given information:
  • Triangle RST is a right triangle with right angle at T
  • L is on side RS, K is on side ST
  • LK is parallel to RT
  • RT = 72 units, LK = 24 units
  • Area of triangle RST = 792 square units
  • Need to find: length of KT

2. INFER what we need to find first

• To use the parallel line properties, we need to know the length of ST
• We can find ST using the area formula since we know the area and one leg (RT)

3. SIMPLIFY to find ST using the area formula

• For right triangle: \(\mathrm{Area} = \frac{1}{2} \times \mathrm{leg_1} \times \mathrm{leg_2}\)
• \(792 = \frac{1}{2} \times 72 \times \mathrm{ST}\)
• \(792 = 36 \times \mathrm{ST}\)
• \(\mathrm{ST} = 792 \div 36 = 22\) units

4. INFER the key geometric relationship

• Since LK is parallel to RT, triangles LKS and RTS are similar
• Both triangles share angle S
• Both have right angles (angle LKS and angle RTS)
• Therefore: \(\frac{\mathrm{LK}}{\mathrm{RT}} = \frac{\mathrm{KS}}{\mathrm{ST}}\) (corresponding sides are proportional)

5. SIMPLIFY using the proportion

• \(\frac{\mathrm{LK}}{\mathrm{RT}} = \frac{\mathrm{KS}}{\mathrm{ST}}\)
• \(\frac{24}{72} = \frac{\mathrm{KS}}{22}\)
• \(\frac{1}{3} = \frac{\mathrm{KS}}{22}\)
• \(\mathrm{KS} = \frac{22}{3}\)

6. INFER how to find KT

• Since K lies on ST: \(\mathrm{ST} = \mathrm{SK} + \mathrm{KT}\)
• Therefore: \(\mathrm{KT} = \mathrm{ST} - \mathrm{SK}\)

7. SIMPLIFY the final calculation

• \(\mathrm{KT} = 22 - \frac{22}{3}\)
• \(\mathrm{KT} = \frac{66}{3} - \frac{22}{3} = \frac{44}{3}\)
• \(\mathrm{KT} = \frac{44}{3} \approx 14.67\) units (use calculator)

Answer: \(\frac{44}{3}\) or \(14.67\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skills: Students don't recognize that LK || RT creates similar triangles, instead trying to use other geometric relationships or formulas inappropriately.

Without recognizing the similar triangles, students may attempt to use Pythagorean theorem directly or try to find relationships that don't exist. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the similar triangles and set up the proportion, but make arithmetic errors when calculating \(\mathrm{KS} = \frac{22}{3}\) or when finding \(\mathrm{KT} = \mathrm{ST} - \mathrm{KS}\).

Common calculation mistakes include getting \(\mathrm{KS} = 22 \times 3\) instead of \(\frac{22}{3}\), or incorrectly subtracting fractions. This may lead them to select incorrect numerical answers.

The Bottom Line:

This problem requires students to see the 'hidden' similar triangles created by the parallel line condition. The key insight is that parallel lines don't just create equal angles—they create entire similar triangles that can be used to find missing lengths through proportions.