Question

If one of the diagonals of a rhombus is equal to one of its sides, then find the greater angle of the rhombus?
(a) 60
(b) 120
(c) 90
(d) 150

Answer 1

Hint: We start solving the problem by drawing the given rhombus and the diagonal. We use the fact that all sides of the rhombus are equal and angles opposite to equal sides in a triangle are equal to get the angle of the triangles which were formed due to the diagonal. We find all the angles in a rhombus and we check which is the greatest angle.

Complete step by step answer:
Given that we had a rhombus which had a length of the diagonal equal to one of its sides. We need to find the greatest of the rhombus.
Let us draw the rhombus ABCD and diagonal equal to its sides to get a better view.

We know that the lengths of the sides of rhombus are equal. Let us assume that the length of the diagonal BD is equal to the sides of the rhombus.
Let us take the triangle BDC and we can see that the length of all the sides of the triangle are equal. We know that angles opposite to the equal sides in a triangle are equal.
So, we have $\angle BCD=\angle CDB=\angle DBC$. We know that the sum of angles in a triangle is ${{180}^{o}}$.
$\Rightarrow \angle BCD+\angle CDB+\angle DBC={{180}^{o}}$.
$\Rightarrow \angle BCD+\angle BCD+\angle BCD={{180}^{o}}$.
$\Rightarrow 3\angle BCD={{180}^{o}}$.
$\Rightarrow \angle BCD=\dfrac{{{180}^{o}}}{3}$.
$\Rightarrow \angle BCD={{60}^{o}}$.
So, we get $\angle BCD=\angle CDB=\angle DBC={{60}^{o}}$ ---(1).
Let us take the triangle ABD and we can see that the length of all the sides of the triangle are equal. We know that angles opposite to the equal sides in a triangle are equal.
So, we have $\angle BAD=\angle ADB=\angle DBA$. We know that the sum of angles in a triangle is ${{180}^{o}}$.
$\Rightarrow \angle BAD+\angle ADB+\angle DBA={{180}^{o}}$.
$\Rightarrow \angle BAD+\angle BAD+\angle BAD={{180}^{o}}$.
$\Rightarrow 3\angle BAD={{180}^{o}}$.
$\Rightarrow \angle BAD=\dfrac{{{180}^{o}}}{3}$.
$\Rightarrow \angle BAD={{60}^{o}}$.
So, we get $\angle BAD=\angle ADB=\angle DBA={{60}^{o}}$ ---(2).
From the figure, we can see that $\angle B=\angle DBA+\angle CBD$.
$\Rightarrow \angle B={{60}^{o}}+{{60}^{o}}$.
$\Rightarrow \angle B={{120}^{o}}$.
Similarly, we can find $\angle D={{120}^{o}}$.
We have found the largest angle as ${{120}^{o}}$.
∴ The largest angle in the given rhombus is ${{120}^{o}}$.

So, the correct answer is “Option B”.

Note: We must remember that all sides in rhombus are equal but we should know that all angles are not equal. If all the angles and sides are equal in a figure, it becomes square which makes all the diagonals equal. The diagonals of the rhombus are not equal. Similarly, we can expect problems to find the length of other diagonals after finding the angles of rhombus.