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Find Area of Triangle

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) = $\frac{1}{2}.ab. Sin C$

area of triangle
Theorem : Prove that the area of triangle ΔABC is given by
Δ = 1/2.bc. SinA = 1/2 ac.SinB = 1/2 ab.Sin C
Proof : Let ABC be a triangle, then there are three cases,
Case I : When ΔABC is an acute triangle :
Sin B = $\frac{AD}{AB}$
=> AD = AB. SinB = c.Sine B
∴ A(ΔABC) = $\frac{1}{2}BC \times AD = \frac{1}{2}.ac. Sin B$

area of acute triangle
Case II : When ΔABC is an obtuse triangle :
Sin (180-B) = $\frac{AD}{AB}$
=> AD = AB. SinB = c.Sine B
∴ A(ΔABC) = $\frac{1}{2}BC \times AD = \frac{1}{2}.ac. Sin B$
area of obtuse triangle
Thus in each case , we have Δ = 1/2 ac.Sin B
Similarly, we can prove that Δ = 1/2.bc. SinA = 1/2 ab.Sin C

Examples on to find area of triangle

1) Find the area of the triangle ABC, in which $\angle$A = 60, b= 4 cm and
c = $\sqrt{3}$ cm.
Solution : The area of ΔABC is given by,
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?
Solution : To know that how much land Jones owns for that we will draw the obtuse triangle ABC, mentioned all the given information and then use the formula to find area of triangle using trigonometry.
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


11th grade math

precalculus

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