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Assertion $A$ : If $A, B, C, D$ are four points on a semi-circular arc with centre at $'O'$ such that $|\overrightarrow{{AB}}|=|\overrightarrow{{BC}}|=|\overrightarrow{{CD}}|$, then $\overrightarrow{{AB}}+\overrightarrow{{AC}}+\overrightarrow{{AD}}=4 \overrightarrow{{AO}}+\overrightarrow{{OB}}+\overrightarrow{{OC}}$
Reason $R$ : Polygon law of vector addition yields $\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C D}+\overrightarrow{A D}=2 \overrightarrow{A O}$
In the light of the above statements, choose the most appropriate answer from the options given below
Both $A$ and $R$ are correct and $R$ is the correct explanation of $A$.
$A$ is not correct but $R$ is correct.
Both $A$ and $R$ are correct but $R$ is not the correct explanation of $A$.
$A$ is correct but $R$ is not correct.
Solution
$|\overrightarrow{A B}|=|\overrightarrow{B C}|=|\overrightarrow{C D}|$
Here, $O$ is the centre of semi- circle
$\therefore|\overrightarrow{O A}|=|\overrightarrow{O B}|=|\overrightarrow{O C}|=|\overrightarrow{O D}|$
Using vector law of addition, we can write,
$\overrightarrow{ AB }=\overrightarrow{ AO }+\overrightarrow{ OB }$
$\overrightarrow{ AC }=\overrightarrow{ AO }+\overrightarrow{ OC }$
$\overrightarrow{ AD }=\overrightarrow{ AO }+\overrightarrow{ OD }=2 \overrightarrow{ AO }$
After adding all, we get,
$\overrightarrow{A B}+\overrightarrow{A C}+\overrightarrow{A D}=4 \overrightarrow{A O}+\overrightarrow{O B}+\overrightarrow{O C}$
Reason $R$ is the direct result of Polygon law of vector addition
Therefore, Polygon law is applicable in both but the equation given in the reason is not useful in explaining the assertion.
