Area of a Triangle and Equation of a Plane JEE Notes | EduRev
Mathematics (Maths) Class 12
JEE : Area of a Triangle and Equation of a Plane JEE Notes | EduRev
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.
Up next
- Examples : Equation of a Plane passing through 3 Non-Collinear points
- Equation of a Plane in Normal Form, Equation of a Plane Perpendicular to a given Vector
- Test: Intercept Form And Family Of Planes
- Intercept Form of the Equation of the Plane
- Equation of a Plane passing through 3 Non-Collinear points and Equation of plane passing through the Intersection of 2 given Planes
- Test: Distance Of A Point From A Plane
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Up next
- Examples : Equation of a Plane passing through 3 Non-Collinear points
- Equation of a Plane in Normal Form, Equation of a Plane Perpendicular to a given Vector
- Test: Intercept Form And Family Of Planes
- Intercept Form of the Equation of the Plane
- Equation of a Plane passing through 3 Non-Collinear points and Equation of plane passing through the Intersection of 2 given Planes
- Test: Distance Of A Point From A Plane
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