Vector Algebra
Vector Operations in Triangles
Grade 12
Question:
<p>If D, E and F are respectively the mid-points of AB, AC and BC in \(\triangle ABC\), then BE + AF is equal to</p>
<p>(a) DC</p>
<p>(b) \(\frac{1}{2}\)BF</p>
<p>(c) 2BF</p>
<p>(d) \(\frac{3}{2}\)BF</p>
Step-by-Step Solution
Key Concept: Express medians as vectors using mid-point formulas and simplify the sum algebraically.
Step 1: Express BE and AF in terms of position vectors: BE = OE - OB AF = OF - OA Step 2: Since E is the mid-point of AC: \(\mathbf{OE} = \frac{\mathbf{OA} + \mathbf{OC}}{2}\) Step 3: Since F is the mid-point of BC: \(\mathbf{OF} = \frac{\mathbf{OB} + \mathbf{OC}}{2}\) Step 4: BE + AF = \(\frac{\mathbf{OA} + \mathbf{OC}}{2} - \mathbf{OB} + \frac{\mathbf{OB} + \mathbf{OC}}{2} - \mathbf{OA}\) Step 5: = \(\frac{\mathbf{OA} + \mathbf{OC} + \mathbf{OB} + \mathbf{OC} - 2\mathbf{OB} - 2\mathbf{OA}}{2}\) Step 6: = \(\frac{2\mathbf{OC} - \mathbf{OA} - \mathbf{OB}}{2} = \mathbf{OC} - \frac{\mathbf{OA} + \mathbf{OB}}{2} = \mathbf{DC}\) ∴ Answer is (a).
Correct Answer: A