Question

$\text{If ABCD is a rhombus, AC is its}\text{smaller diagonal and}\angle ABC={60}^{\circ },\text{then find the length of a side of the}\text{rhombus when}AC=7cm.$

• A. 6 cm
• B. 6.5 cm
• C. 7 cm
• D. 7.5 cm

Answer 1

The correct option is C 7 cm

$\text{Given:}\angle ABC={60}^{\circ }\text{and AC = 7 cm}$
$\text{In rhombus ABCD,}$


$AB=BC\text{(In a rhombus all sides are equal.)}$
$\therefore △ABC\text{is isosceles and base angles}\text{are equal.}$

$\text{Given:}\angle ABC={60}^o$

$\because \text{Rhombus is a parallelogram and in}$$\text{a parallelogram, the adjacent angles}\text{are supplementary.}$

$\Rightarrow \angle ABC+\angle BCD={180}^{\circ }\Rightarrow {60}^{\circ }+\angle BCD={180}^{\circ }\Rightarrow \angle BCD={180}^o-{60}^o\Rightarrow \angle BCD={120}^o$

$\because \text{Diagonals in rhombus bisect}$
$\text{the angles.}$
$\therefore \angle BCA=\angle CAB=\frac{{120}^{\circ }}{2}={60}^{\circ }$
$\therefore △ABC\text{is an equilateral triangle.}$

$\text{So, AC = AB = BC = 7 cm}\text{Therefore, length of a side of the}\text{rhombus is 7 cm.}$