Question

Two sides of a triangle are 7 and 6 centimetres long and the angle between them is 140°. What is the length of the third side?

Answer 1

Let the given triangle be ABC with AB = 6 cm, BC = 7 cm and ∠ABC = 140°.

Construction: Produce CB and draw AD perpendicular to CB.

By construction, we have ∠ADB = 90°

We know that the sum of angles of a linear pair is 180°.

∴∠ABC + ∠ABD = 180°

⇒ 140° + ∠ABD = 180°

⇒ ∠ABD = 180° − 140° = 40°

In ΔABC:

⇒ AD = AB × Sin 40° … (1)

We know that Sin 40° ≈ 0.6428 (From table)

Putting the value in (1):

AD ≈ 6 × 0.6428

= 3.8568 cm

⇒ DB = AB × Cos 40° … (2)

We know that Cos 40° ≈ 0.7660 (from table)

Putting the value in equation (2):

DB ≈ 6 × 0.7660 cm

= 4.596 cm

DC = DB + BC

≈ (4.596 + 7) cm

= 11.596 cm

Now, in ΔADC:

Thus, the length of the third side of the triangle is approximately equal to12.22 cm.