Area of a Triangle and Equation of a Plane | Mathematics (Maths) Class 12 - JEE PDF Download

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

∴ the d.c.’ s of OA are

and the d.c.’s of OB are

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

sin θ


Hence the area of ΔOAB

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

Sol. Let Δx, Δy, Δz be the areas of the projections of the area Δ of triangle ABC on the yz, zx and xy-planes respectively. We have

Δx =

Δy =

Δz =

∴ the required area Δ =

Ex.7 A plane is passing through a point P(a, –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, –2a, 2a) is
A(x – a) + B(y + 2a) + C(z – 2a) = 0.
∵ the direction cosines of the normal OP to the plane ABC are proportional to a – 0, –2a – 0, 2a – 0 i.e. a, –2a, 2a. ⇒ equation of plane ABC is

a(x – a) – 2a(y + 2a) + 2a(z – 2a) = 0 or ax – 2ay + 2az = 9a² ....(1)

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