Question : If ABCD is a rhombus, AC is its smallest diagonal, and $\angle$ABC = $60^{\circ}$, find the length of a side of the rhombus when AC = 6 cm.
Option 1: $6$ cm
Option 2: $3$ cm
Option 3: $6\sqrt2$ cm
Option 4: $3\sqrt3$ cm
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Correct Answer: $6$ cm
Solution :
In rhombus ABCD, AB = BC = CD = DA
AC = 6 cm
$\angle$ABC = $60^{\circ}$
In $\triangle$ABC,
AB = BC
So, $\angle$BAC = $\angle$BCA
Now, $\angle$BAC + $\angle$BCA + $\angle$ABC = $180^{\circ}$
⇒ $\angle$BAC + $\angle$BCA = $120^{\circ}$
⇒ $\angle$BAC = $\angle$BCA = $\angle$ABC = $60^{\circ}$
⇒ $\triangle$ABC is an equilateral triangle.
⇒ AB = BC = AC = 6 cm
Hence, the correct answer is $6$ cm.
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