Question

Two sides of a parallelogram are of length 6 centimetres and 4 centimetres and the angle between them is 35°. What are the lengths of its diagonals?

Answer 1

Let the given parallelogram be ABCD with DC = 6 cm, AD = 4 cm and ∠ADC = 35°.

Construction: Draw AE perpendicular to DC and join AC.

In ΔAED:

⇒ AE = AD × Sin 35° … (1)

We know that Sin 35° ≈ 0.5736 (From table)

Putting the value in (1):

AE ≈ 4 × 0.5736 cm

= 2.2944 cm

⇒ DE = AD × cos 35° … (2)

We know that cos 35° ≈ 0.8192 (From table)

Putting the value in (2):

DE ≈ 4 × 0.8192 cm

= 3.2768 cm

EC = DC − DE

≈ (6 − 3.2768) cm

= 2.7232 cm

Now, in ΔAEC:

Now, consider the parallelogram again.

Construction: Produce DC and draw BF perpendicular to DC. Join BD.

As AD is parallel to BC, ∠ADC = ∠BCF = 35°.

Also, AE = BF = 2.2944 cm

In ΔBFC:

⇒ CF = BC × cos 35° … (2)

We know that cos 35° ≈ 0.8192 (From table)

Putting the value in (2):

CF ≈ 4 × 0.8192 cm

= 3.2768 cm

DF = DC + CF

≈ (6 + 3.2768) cm

= 9.2768 cm

Now, in ΔBFD:

Thus, the length of the diagonals of the given parallelogram is 3.56 cm and 9.56 cm.