Vector Algebra
Angle Bisector
Grade 12
Question:
<p>If internal and external bisectors of ∠A of △ABC meet the base BC at D and E respectively (D and E lie on same side of B), then which relation holds?</p>
<p>(a) \(BC = \frac{BD + BE}{4}\)</p>
<p>(b) \(BC = BD \times DE\)</p>
<p>(c) \(\frac{2}{BC} = \frac{1}{BD} + \frac{1}{BE}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Internal and external angle bisectors divide the opposite side in the ratio of adjacent sides; this creates a harmonic relationship between BD, BE, and BC.
Step 1: By angle bisector theorem, the internal bisector divides BC in ratio AB : AC, so D divides BC in ratio AB : AC.
Step 2: The external bisector divides BC externally in ratio AB : AC, so E divides BC externally.
Step 3: If D divides BC internally in ratio m : n and E divides BC externally in ratio m : n, then BD and BE are related such that \(\frac{1}{BD} + \frac{1}{BE} = \frac{2}{BC}\).
∴ Answer is C.
Correct Answer: c