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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 <img alt="\begin{aligned} &|\overrightarrow{\mathrm{AB}}|=|\overrightarrow{\mathrm{BC}}|=|\overrightarrow{\mathrm{CD}}| \t

Assertion A : If A , B, C, D are four points on a semi-circular arc with centre at 'O' such that
\begin{aligned} &|\overrightarrow{\mathrm{AB}}|=|\overrightarrow{\mathrm{BC}}|=|\overrightarrow{\mathrm{CD}}| \text {, then } \\ &\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{AC}}+\overrightarrow{\mathrm{AD}}=4 \overrightarrow{\mathrm{AO}}+\overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}} \end{aligned}
Reason R : Polygon law of vector addition yields
\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}+\overrightarrow{\mathrm{CD}}+\overrightarrow{\mathrm{AD}}=2 \overrightarrow{\mathrm{AO}}\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}+\overrightarrow{\mathrm{CD}}+\overrightarrow{\mathrm{AD}}=2 \overrightarrow{\mathrm{AO}} In the light of the above statements, choose the most appropriate answer from the options given below:
Option: 1 A is correct but R is not correct.
Option: 2 A is not correct but R is correct.
Option: 3 Both A and R are correct and R is the correct explanation of A.
Option: 4 Both A and R are correct but R is not the correct explanation of A.

Answers (1)

best_answer

Assertion : |\overline{A B}|=|\overline{B C}|=(\overline{C D})

\begin{aligned} \overline{A B}+\overline{A C}+\overline{A D} &=\overline{A B}+(\overline{A B}+\overline{B C})+(\overline{A B}+\overline{B C}+\overline{C D}) \\ &=3 \overline{A B}+2 \overline{B C}+\overline{C D} \\ &=\overline{A B}+2 ( \overline{A B}+\overline{B C})+\overline{C D} \\ &=\overline{A B}+2 \overline{A C}+\overline{C D} \end{aligned}

\begin{aligned} &\overline{A B}=\overline{A O}+\overline{O B} \\ &\overline{A C}=\overline{A O}+\overline{O C} \end{aligned}

\overline{A O}= \overline{O D}=\overline{O C}+\overline{C D}

                       \begin{aligned} &=\overline{A O}+\overline{O B}+2 \overline{A O}+2 \overline{O C}+\overline{C D} \\ &=3 \overline{A O}+\overline{O B}+\overline{O C}+\overline{O C}+\overline{CD} \\ &=4\overline{AO}+\overline{O B}+\overline{O C} \end{aligned}

\therefore \overline{A B}+\overline{A C}+\overline{A D}=\overline{4 A O}+\overline{O B}+\overline{O C}

Reason : LHS : \overline{A B}+\overline{B C}+\overline{C D}+\overline{A D}

By polyon law,

\overline{A B}+\overline{B C}+\overline{C D}=\overline{A D}=2 \overline{A O}

\therefore LHS= RHS

The reason is not correct for the assertion but both the statements are true.

The correct option is (4)

Posted by

vishal kumar

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