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
∴ 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
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
∴ 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.
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.