In triangle ABC, the measure of angle B is 90° and BD is an altitude of the triangle. The length...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC has a right angle at B (\(\mathrm{angle\:B = 90°}\))
  • BD is an altitude of the triangle
  • \(\mathrm{AB = 15}\)
  • AC is 23 greater than AB, so \(\mathrm{AC = 15 + 23 = 38}\)
  • Need to find: \(\mathrm{BC/BD}\)

2. INFER what the altitude creates

• Since BD is an altitude from vertex B to side AC, BD is perpendicular to AC
• This means \(\mathrm{angle\:BDC = 90°}\)
• We now have two triangles with right angles: the original triangle ABC (right angle at B) and triangle BDC (right angle at D)

3. INFER the similar triangles

• Triangle ABC and Triangle BDC are similar by AA similarity because:
  • Angle C is shared by both triangles
  • \(\mathrm{Angle\:ABC = Angle\:BDC = 90°}\)
• When triangles are similar, their corresponding sides are proportional

4. INFER the correct proportion

• From \(\mathrm{Triangle\:ABC \sim Triangle\:BDC}\), the corresponding sides give us:
  \(\mathrm{AC/AB = BC/BD}\)
• This is the key relationship we need!

5. SIMPLIFY to find the answer

• Substitute the known values into \(\mathrm{AC/AB = BC/BD}\):
  \(\mathrm{38/15 = BC/BD}\)
• Therefore: \(\mathrm{BC/BD = 38/15}\)

Answer: D. \(\mathrm{38/15}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize that similar triangles are involved but set up the wrong proportion. They might think that since we're looking for \(\mathrm{BC/BD}\), they should use the relationship \(\mathrm{AC/BC = BC/BD}\) (from the geometric mean relationship in right triangles). This leads them down a complex calculation path involving finding BC first using the Pythagorean theorem, then finding BD, which results in messy calculations that don't match any answer choice. This causes them to get stuck and randomly select an answer.

Second Most Common Error:

Missing conceptual knowledge about altitudes: Students don't recognize that BD being an altitude means BD is perpendicular to AC, so they don't identify that \(\mathrm{angle\:BDC = 90°}\). Without this key insight, they can't establish the similar triangles needed for the solution. This leads to confusion and guessing.

The Bottom Line:

This problem requires recognizing that an altitude in a triangle creates similar triangles, then correctly identifying which sides correspond in the similarity relationship. The beauty is that you don't need to calculate individual side lengths - the proportion gives you the answer directly.