Calculate the area of the triangle shown.

Let ABC be the required triangle.
Draw perpendicular from AS on BC.

Let AS = x cm.

In Δ BAS,

∠ABS + ∠BAS + ∠ASB = 180°
⇒ 45° + ∠BAS + 90° = 180°
⇒ ∠BAS = 45°
⇒ Δ BAS is an isosceles triangle.

∴ AS = BS = x cm

BC = BS + CS = 4 cm
⇒ x + CS = 4 cm
⇒ CS = (4–x) cm

In Δ CAS,

∠ACS + ∠CAS + ∠ASC = 180°
⇒ 60° + ∠CAS + 90° = 180°
⇒ ∠CAS = 30°

We know that sides of any triangle of angles 30°, 60° and 90°
are in the ratio 1: √3: 2 .

⇒ CS :AS: AC = 1: √3: 2
⇒ (4-x) :x : AC = 1: √3 :2
⇒ (4 - x) : x = 1 : √3
⇒ √3(4 - x) = x
⇒ 4√3 - x√3 = x
⇒ x(√3 + 1) = 4√3

Multiplying and Dividing by (√3 - 1)

⇒ x = 2(√3 - 1) cm

In Δ BAC,

Height = x = 2(√3 - 1) cm and Base = 4 cm

Area (Δ) = 4(√3 - 1) cm² = 10.92 cm²