• # question_answer Which one of the following is not true for the equilateral triangle ABC given below? A)  $\frac{\text{AB}+\text{AC}}{2}=\text{BC}$ B) $\angle \text{BAC}+\angle \text{ACB}=120{}^\circ$ C) $\frac{\text{AB}}{\text{AC}}=\frac{\text{AC}}{\text{BC}}$ D)  $\frac{\angle \text{BAC}+\angle \text{ACB}+\angle \text{ABC}}{2}=2\times 90{}^\circ$ E) None of these

Correct Answer: D

Solution :

Explanation: Option (d) is correct. Since, $\vartriangle \text{ABC}$ is an equilateral triangle, therefore, $\angle \text{ABC}=\angle \text{ACB}=\angle \text{BAC}=60{}^\circ$ (each) and, AB = BC = AC = 1 unit (say) Now, in option (a); $\frac{\text{AB+AC}}{2}=\frac{1\text{unit}+1\text{unit}}{2}$ $=\frac{2\text{units}}{2}=1\text{unit=BC}$ In option (b); $\angle \text{BAC}+\angle \text{ACB}=60{}^\circ +60{}^\circ =120{}^\circ$ In option (c); $\frac{\text{AB}}{\text{BC}}=\frac{1\text{unit}}{1\text{unit}}=1;=\frac{\text{AC}}{\text{BC}}=\frac{1\text{unit}}{1\text{unit}}=1$ In option (d); $\frac{\angle \text{BAC}+\angle \text{ACB}+\angle \text{ABC}}{2}$ $=\frac{60{}^\circ +60{}^\circ +60{}^\circ }{2}=\frac{180{}^\circ }{2}=90{}^\circ$ $=1\times 90{}^\circ =1$ right angle (not 2 right angles).

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