# Calculate the are

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

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