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Question

Using integration, find the area of the triangle whose vertices are (2,3),(3,5), and (4,4).

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Solution

Equation of the line passing through points (2,3) and (3,5) is
y3x2=5332y3=2x4y2x+1=0 (1)

Equation of the line passing through points (4,4) and (2,3) is
y3x2=43422y6=x22yx4=0 (2)

Equation of the line passing through points (4,4) and (3,5) is
y5x3=4543y5=3xy+x8=0(3)

Solving (1) & (2), we get the coordinates x=2, y=3.
Solving (2) & (3), we get the coordinates x=4, y=4.
Solving (3) & (1), we get the coordinates x=3, y=5.

Graph of the above equations will be
Required area is
32(2x1) dx+43(8x) dx42(x+42)dx=[(2x1)22×2]32+[(8x)22]43[(x+4)24]42=14[(5232)2(4252)(8262)]=32 sq. units

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