Question

Find lengths of diagonals AC and BD. Given AB = 24 cm and ∠BAD = 60°.

Answer 1

The given figure is a rhombus as all sides are equal. we know that diagonals of a rhombus bisect each other at right angles and also bisect the angle of vertex.
Let the diagonals AC and BD intersect each other at O.

⇒ OA = `"OC" - (1)/(2)"AC", "OB" = "OD" = (1)/(2)"BD"`, ∠AOB = 90°

Now, ∠BAD = 60°

⇒ ∠OAB = `(1)/(2)∠"BAD"` = 30°

In right-angled AOB,

sin30° = `"OB"/"AB"`

⇒ `(1)/(2) = "OB"/(24)`
⇒ OB = 12cm

cos30° = `"OA"/"AB"`

⇒ `sqrt(3)/(2) = "OA"/(24)`
⇒ OA = `12sqrt(3)"cm"`
∴ Length of diagonal AC
= 2 x OA
= `2 xx 2sqrt(3)`
= `24sqrt(3)"cm"`
And, length of diagonal BD
= 2 x OB
= 2 x 12
= 24cm.