Question

The measure of each of the two opposite angles of a rhombus is ${60^o}$ and the measure of one of its sides is 10 cm. The length of its smaller diagonal is:
(A) 10 cm
(B) $10\sqrt 3 $ cm
(C) $10\sqrt 2 $ cm
(D) $\dfrac{{5\sqrt 2 }}{2}$ cm

Answer 1

Hint:- We will use the properties of rhombus which states that all the sides are equal to each other and the diagonals bisect the angles of the rhombus through which they pass to find out the angles of the triangle formed by the shorter diagonal

Complete step-by-step answer:
Let ABCD be the given rhombus. The angles ADC and ABC be the angles whose measure is given to be equal to ${60^o}$.
Anyone side of the rhombus ABCD is given to be 10 cm.
But we know that all the sides of a rhombus are equal to each other.
AB = BC = CD = AD = 10 cm

Let us now focus on finding out which type of triangle is triangle ADC.
From the properties of rhombus we know that $\angle BAD + \angle CDA = {180^o}$ (because when a transversal AD intersects two parallel lines AB and CD, the interior angles on the same side of the transversal are supplementary)
Thus, we get
$\angle BAD = {180^o} - \angle CDA = {180^o} - {60^o} = {120^o}$
Similarly, from the properties of rhombus we know that $\angle BCD + \angle CDA = {180^o}$ (because when a transversal CD intersects two parallel lines AD and BC, the interior angles on the same side of the transversal are supplementary)
Thus, we get
$\angle BCD = {180^o} - \angle CDA = {180^o} - {60^o} = {120^o}$
We know that the diagonals of a rhombus bisect the angles of the rhombus.
The diagonal AC bisects the angles BCD and BAD.
Hence, we get $\angle DAC = \dfrac{1}{2}\angle BAD = \dfrac{1}{2} \times {120^o} = {60^o}$ and
$\angle DCA = \dfrac{1}{2}\angle BCD = \dfrac{1}{2} \times {120^o} = {60^o}$
Now we can see that all the angles of triangle ACD are equal to ${60^o}$ , that is triangle ACD is an equilateral triangle.
Now, from the definition of equilateral triangle, all the sides of the triangle are of equal length.
$\therefore AD = AC = CD = 10\;cm$
From the figure of the rhombus it is evident that AC is the shorter diagonal and we just found out that its length is equal to 10 cm.

Hence, the correct option is (A)

Note:- The first property implies that every rhombus is a parallelogram. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel, adjacent angles are supplementary, the two diagonals bisect one another, any line through the midpoint bisects the area, and the sum of the squares of the sides equals the sum of the squares of the diagonals.