Triangle ABC with side lengths AB = 3 cm, AC = 5 cm, BC = 6 cm. If AD is the internal angle bisector of ∠A meeting BC at D, find BD.

Introduction / Context:The Angle Bisector Theorem relates the division of the opposite side to the adjacent sides. It is frequently used to split a side into two segments proportional to the neighbouring side lengths.

Given Data / Assumptions:

• |AB| = 3 cm, |AC| = 5 cm, |BC| = 6 cm.
• AD bisects ∠A internally, meeting BC at D.

Concept / Approach:Angle Bisector Theorem: BD/DC = AB/AC. Also BD + DC = BC. Solve the proportion to find BD explicitly.

Step-by-Step Solution:Let BD = x, then DC = 6 − x.x / (6 − x) = AB/AC = 3/5.Cross-multiply: 5x = 18 − 3x ⇒ 8x = 18 ⇒ x = 18/8 = 2.25 cm.

Verification / Alternative check:Check ratio: BD:DC = 2.25 : 3.75 = 3 : 5 (correct).

Why Other Options Are Wrong:2, 2.5, and 3 cm do not maintain BD/DC = 3/5 with BC fixed at 6 cm.

Common Pitfalls:Using AB/AC = BD/BC by mistake; the theorem relates BD/DC, not BD/BC.

Final Answer:2.25 cm