In the figure, ∠BAD = ∠CAD (AD is the angle bisector at A). Given AB = 4 cm, AC = 5.2 cm, and BD = 3 cm. Find BC.

Introduction / Context:With AD the internal bisector of ∠A, the Angle Bisector Theorem gives a proportional split of side BC. From one segment (BD) we can obtain the other (DC) and thus BC.

Given Data / Assumptions:

• AB = 4, AC = 5.2.
• BD = 3, need BC.

Concept / Approach:Angle Bisector Theorem: BD/DC = AB/AC ⇒ DC = BD * (AC/AB). Then BC = BD + DC.

Step-by-Step Solution:

Verification / Alternative check:BD:DC = 3:3.9 = 30:39 = 10:13, and AB:AC = 4:5.2 = 40:52 = 10:13 ✓.

Why Other Options Are Wrong:9.6 and 9.3 add incorrect DC values; 3.9 is only DC, not BC.

Common Pitfalls:Inverting AB/AC; forgetting to add BD to get BC; rounding prematurely.

Final Answer:6.9 cm