In right triangle ABC, the right angle is at vertex B. Point D is on AC and point E is...

1. TRANSLATE the given information

• Given information:
  • Right triangle ABC with right angle at B
  • DE is parallel to AB
  • BC = 30, EC = 10
  • Area of triangle ABC = 360 square units
  • Need to find: length of DE

2. INFER what we need first

• To use similarity properties, we need to know the length of AB
• We can find AB using the area formula since we know the area and one leg (BC)

3. TRANSLATE the area information and SIMPLIFY

• Area formula for right triangle: \(\mathrm{Area} = \frac{1}{2} \times \mathrm{base} \times \mathrm{height}\)
• \(360 = \frac{1}{2} \times \mathrm{AB} \times 30\)
• \(360 = 15 \times \mathrm{AB}\)
• \(\mathrm{AB} = 24\)

4. INFER the key similarity relationship

• Since DE is parallel to AB, triangles CDE and CAB are similar
• This happens because angle C is common to both triangles, and the parallel lines create equal corresponding angles (AA similarity)

5. TRANSLATE similarity into a proportion and SIMPLIFY

• For similar triangles, corresponding sides are proportional:
• \(\frac{\mathrm{DE}}{\mathrm{AB}} = \frac{\mathrm{EC}}{\mathrm{BC}}\)
• \(\frac{\mathrm{DE}}{24} = \frac{10}{30}\)
• \(\frac{\mathrm{DE}}{24} = \frac{1}{3}\)
• \(\mathrm{DE} = 24 \times \frac{1}{3} = 8\)

Answer: B) 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that DE || AB creates similar triangles. Instead, they try to use the Pythagorean theorem or other geometric relationships that don't directly apply. Without recognizing the similarity, they get stuck and cannot establish the key proportion needed to find DE. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the similar triangles and set up the proportion DE/AB = EC/BC, but make computational errors. They might incorrectly calculate AB from the area (getting something other than 24), or make arithmetic mistakes when solving DE/24 = 10/30. These calculation errors often lead them to select Choice A (6) or Choice C (12).

The Bottom Line:

This problem requires students to see the "big picture" connection between parallel lines and similar triangles, then execute multiple computational steps accurately. The geometric insight is crucial - without recognizing similarity, the numerical information becomes disconnected and unusable.