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**D. AREA OF A TRIANGLE**

Show that the area of a triangle whose vertices are the origin and the points and

The direction ratios of OA are

Also OA

and OB

âˆ´ the d.c.â€™ s of OA are

and the d.c.â€™s of OB are

Hence if Î¸ is the angle between the line OA and OB, then

sin Î¸

Hence the area of Î”OAB

**Ex.6 Find the area of the triangle whose vertices are A(1, 2, 3), B(2, â€“1, 1)and C(1, 2, â€“4).**

**Sol. **Let Î”x, Î”y, Î”z be the areas of the projections of the area Î” of triangle ABC on the yz, zx and xy-planes respectively. We have

Î”x =

Î”y =

Î”z =

âˆ´ the required area Î” = **Ex.7 A plane is passing through a point P(a, â€“2a, 2a), **** at right angle to OP, where O is the origin to meet the axes in A, B and C. Find the area of the triangle ABC.**

**Sol.** OP

Equation of plane passing through P(a, â€“2a, 2a) is

A(x â€“ a) + B(y + 2a) + C(z â€“ 2a) = 0.

âˆµ the direction cosines of the normal OP to the plane ABC are proportional to a â€“ 0, â€“2a â€“ 0, 2a â€“ 0 i.e. a, â€“2a, 2a. â‡’ equation of plane ABC is

a(x â€“ a) â€“ 2a(y + 2a) + 2a(z â€“ 2a) = 0 or ax â€“ 2ay + 2az = 9a^{2} ....(1)

Now projection of area of triangle ABC on ZX, XY and YZ planes are the triangles AOC, AOB and BOC respectively.

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