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  • Explain solution RD Sharma class 12 chapter Scaler and Dot Product exercise 23.2 question 8 maths

Explain solution RD Sharma class 12 chapter Scaler and Dot Product exercise 23.2 question 8 maths

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Answer: A B^{2}+A C^{2}=2\left(A D^{2}+C D^{2}\right)

Hint: Use Pythagoras theorem

Given: \text { if AD is median of } \triangle A B C \text { using vector }

\text { P.T } A B^{2}+A C^{2}=2\left(A D^{2}+C D^{2}\right)

Solution: A D \text { is median of } \triangle \mathrm{ABC}

Draw \mathrm{AE} \perp \mathrm{BC}

In right angled \Delta ACE

(A B)^{2}=(A E)^{2}+(B E)^{2}            (By Pythagoras Theorem)

In right angled triangle \Delta ACE

(A C)^{2}=(A E)^{2}+(E C)^{2}

Add (1) & (2)

\begin{aligned} &(A B)^{2}+(A C)^{2}=(A E)^{2}+(B E)^{2}+(A E)^{2}+(E C)^{2} \\\\ &(A B)^{2}+(A C)^{2}=2(A E)^{2}+(B D-E D)^{2}+(E D+D C)^{2} \end{aligned}

\begin{aligned} &(A B)^{2}+(A C)^{2}=2(A E)^{2}+(B D)^{2}-2 B D \cdot E D+(E D)^{2}+(E D)^{2}+2 E D \cdot D C+(D C)^{2} \\\\ &(A B)^{2}+(A C)^{2}=2(A E)^{2}+2(E D)^{2}+(D C)^{2} \end{aligned}

\begin{aligned} &(A B)^{2}+(A C)^{2}=2(A E)^{2}+2(E D)^{2}+2(B D)^{2} \quad(B D=D C) \\\\ &(A B)^{2}+(A C)^{2}=2\left((A E)^{2}+(E D)^{2}+(B D)^{2}\right) \\\\ &(A B)^{2}+(A C)^{2}=2\left(A D^{2}+C D^{2}\right) \end{aligned}

 

 

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