In a \(\triangle A B C\) (shown in the figure below). State whether the following are true or false (i)…

In a \(\triangle A B C\) (shown in the figure below). State whether the following are true or false
(i) \(\mathbf{A B}+\mathbf{B C}+\mathbf{C A}=\mathbf{0}\) (ii) \(\mathbf{A B}+\mathbf{B C}-\mathbf{A C}=\mathbf{0}\) (iii) \(\mathbf{A B}-\mathbf{C B}+\mathbf{C A}=\mathbf{0}\) (iv) \(\mathbf{A B}+\mathbf{B C}-\mathbf{C A}=\mathbf{0}\)

1. (i) True (ii) True (iii) True (iv) False
2. (i) True (ii) False (iii) True, (iv) False
3. (i) False (ii) False (iii) True (iv) True
4. (i) False (ii) True (iii) False (iv) False

Solution

Following triangle came, \(\begin{array}{llll} & \mathbf{A B}+\mathbf{B C} =\mathbf{A C} \\ \Rightarrow & \mathbf{A B}+\mathbf{B C}+\mathbf{C A} =0 \quad \ldots (i) \\ \text {and } & \mathbf{A B}+\mathbf{B C}-\mathbf{A C} =0 \quad \ldots (ii) \\ \text {Also, } & \mathbf{C A}+\mathbf{A B} =\mathbf{C B} \\ \Rightarrow & \mathbf{C A}+\mathbf{A B}-\mathbf{C B} =0 \\ \text {or } & \mathbf{A B}-\mathbf{C B}+\mathbf{C A} =0 \quad \ldots (iii) \end{array}\)

Asked in: AP EAMCET 2020 (17 Sep Shift 2)

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