Question

According to Apolloneous Theorem, if $\overline{A}D$ is a median of $△ABC$, then $A{B}^2+A{C}^2=$

• A. $AD+BD$
• B. $AD-BD$
• C. $2(A{D}^2+B{D}^2)$
• D. $2(A{D}^2-B{D}^2)$

Answer 1

The correct option is C $2(A{D}^2+B{D}^2)$

$\begin{array}{cc}Then,A{B}^2+A{C}^2=?& \\ ADismedian& \\ \Rightarrow A{B}^2+A{C}^2=2\left|\frac{B{C}^2}{4}\right|+2A{D}^2& \left[\begin{array}{c}BC=2BD\\ orBC=DC\end{array}\right]\\ \Rightarrow A{B}^2+A{C}^2=2{\left|\frac{B{C}^{}}{2}\right|}^2+2A{D}^2& \\ A{B}^2+A{C}^2=2B{D}^2+2A{D}^2=2\left(A{D}^2+B{D}^2\right)& \end{array}$