Question

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In a rhombus ABCD, angle A = 60°, AB = 10 cm.
Find the length of the diagonals.​

Answer 1

Given :

ABCD is a rhombus with ∠A = 60°, AB = 10 cm.

To Find:

We have to find the length of the diagonals, AC and BD.

Solution :

The important properties of the diagonals of a rhombus are:

1. the diagonals of a rhombus bisect each other
2. the diagonals are perpendicular to each other
3. the diagonals bisect the angles of the rhombus

Let O be the point where the two diagonals intersect each other as shown in the figure.

Given that ∠BAD = 60°,

By using the diagonal properties, ∠BAO= 30° and ∠AOB= 90°

∴ By using the properties of the interior angles of a triangle,

                  ∠ABO+∠BAO+∠AOB= 180°

                  ∠ABO= 180°-30°-90°

⇒                ∠ABO =60°

In right-angled ΔAOB,

                     $\[\sin {30^ \circ } = \frac{1}{2}\]$

                        $\[\begin{array}{c}\frac{{OB}}{{AB}} = \frac{1}{2}\\\\\frac{{OB}}{{10}} = \frac{1}{2}\\\\\therefore OB = 5\end{array}\]$

Similarly, we have,    

                       $\[\cos {30^ \circ } = \frac{{\sqrt 3 }}{2}\]$

By using the diagonal properties, AO= OC and OB= OD

⇒ Length of AC = 2×AO = 10√3

⇒ Length of BD = 2×OB = 10

Hence, the length of the diagonals is 10 and 10√3.