Which one of the following is correct?
Consider the following for the next two (02) items that follow: ABC is triangle in which AB=AC and D is any point on BC.
- A. AB^{2}-AD^{2}=AD\times BD
- B. AC^{2}-AD^{2}=BD\times CD ✓
- C. AB^{2}-AD^{2}=2AD\times BD
- D. AC^{2}-AD^{2}=2BD\times CD
Correct Answer: B. AC^{2}-AD^{2}=BD\times CD
Explanation
By Stewart's Theorem on \triangle ABC with AB=AC, we have AB^2 \cdot CD + AC^2 \cdot BD = BC(AD^2 + BD \cdot CD). Since AB=AC and BC = BD+CD, this simplifies to AC^2(BD+CD) = (BD+CD)(AD^2 + BD \cdot CD) \implies AC^2 = AD^2 + BD \cdot CD \implies AC^2 - AD^2 = BD \times CD.
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