When two sides and angle included between (SAS)them is given then there is another formula to find area of triangle. There are three equivalent formulas based on the sides and angle given. The area of triangle ABC is given by, • A(ΔABC) = $\frac{1}{2}.bc. Sin A$ •A(ΔABC) = $\frac{1}{2}.ac. Sin B$ •A(ΔABC) = $\frac{1}{2}.ab. Sin C$ |

Δ = 1/2.bc. SinA = 1/2 ac.SinB = 1/2 ab.Sin C

Case I : When ΔABC is an acute triangle : Sin B = $\frac{AD}{AB}$ => AD = AB. SinB = c.Sine B ∴ A(ΔABC) = $\frac{1}{2}BC \times AD = \frac{1}{2}.ac. Sin B$ |

Case II : When ΔABC is an obtuse triangle : Sin (180-B) = $\frac{AD}{AB}$ => AD = AB. SinB = c.Sine B ∴ A(ΔABC) = $\frac{1}{2}BC \times AD = \frac{1}{2}.ac. Sin B$ |

Similarly, we can prove that Δ = 1/2.bc. SinA = 1/2 ab.Sin C

c = $\sqrt{3}$ cm.

A(ΔABC) = $\frac{1}{2}.bc. Sin A$

= $\frac{1}{2}.4\sqrt{3}. Sin 60$

= $2\sqrt{3}. \frac{\sqrt{3}}{2}$

A(ΔABC) = 3 sq.cm

2) Farmer Jones owns a triangular piece of land.The length of the fence AB is 120 m. The length of the fence BC is 230 m. The angle between fence AB and fence BC is 125º.How much land does Farmer Jones own?

AB = c = 120 m

BC = a = 230 m

$\angle$ B = 125$^{0}$

The area of ΔABC is given by,

A(ΔABC) = $\frac{1}{2}.ac. Sin B$

= $\frac{1}{2}.(120).(230). Sin 125$

= 13800. (0.819152)

A(ΔABC) = 11304.2976 sq.cm

A(ΔABC) = 11304.30 sq.cm

precalculus

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