Question

How many rhombuses can we draw with one diagonal 5 centimetres long and one angle 50°? What are their areas?

Answer 1

We can draw two rhombuses with one diagonal 5 cm long and one angle 50°, explained in the following cases.

Case I:

Let ABCD be a rhombus with AC = 5 cm, ∠ADC = 50°.

Let the diagonals AC and BD intersect each other at point O.

We know that diagonals of a rhombus are perpendicular bisectors of each other.

Also, diagonals of a rhombus bisect the vertex angles.

∴ ∠DOC = 90°

OC =

∠CDO =

In ΔDOC:

⇒ DO = … (1)

We know that tan 25° ≈ 0.4663 (From table)

Putting the value in (1):

DO ≈ cm

BD = 2 × DO

≈ 2 × 5.36 cm

= 10.72 cm

Thus, with the length of both the diagonals and measure of one angle, we can draw a unique rhombus.

Area of the rhombus =

Case 2:

Let ABCD be a rhombus with BD = 5 cm, ∠ADC = 50°.

Let the diagonals AC and BD intersect each other at point O.

We know that diagonals of a rhombus are perpendicular bisectors of each other.

Also, diagonals of a rhombus bisect the vertex angles.

∴ ∠DOC = 90°

OD =

∠CDO =

In ΔDOC:

⇒ OC = DO × tan 25° … (1)

We know that tan 25° ≈ 0.4663 (From table)

Putting the value in (1):

⇒ OC ≈ 2.5 × 0.4663 cm

= 1.17 cm

AC = 2 × OC

≈ 2 × 1.17 cm

= 2.34 cm

Thus, with the length of both the diagonals and measure of one angle, we can draw a unique rhombus.

Area of the rhombus =