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Area of a Triangle and Equation of a Plane - CUET Preparation - Commerce
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.
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FAQs on Area of a Triangle and Equation of a Plane
| 1. What is the formula for finding the area of a triangle? |  |
Ans. The formula for finding the area of a triangle is (base * height) / 2. This means you multiply the base of the triangle by its height and then divide the result by 2.
| 2. How do you calculate the area of a triangle if you know the lengths of its sides? | |
Ans. If you know the lengths of the sides of a triangle, you can calculate its area using Heron's formula. Heron's formula states that the area of a triangle with sides of lengths a, b, and c can be found using the formula: area = sqrt(s * (s-a) * (s-b) * (s-c)), where s is the semi-perimeter of the triangle (s = (a+b+c)/2).
| 3. Can you find the area of a triangle if you know only the lengths of two sides and the included angle? | |
Ans. Yes, you can find the area of a triangle if you know the lengths of two sides and the included angle using the formula: area = (1/2) * a * b * sin(C), where a and b are the lengths of the two sides and C is the included angle.
| 4. How can you determine the equation of a plane given three non-collinear points on the plane? | |
Ans. To determine the equation of a plane given three non-collinear points on the plane, you can use the point-normal form of the equation. First, calculate the normal vector by finding the cross product of two vectors formed by the three points. Then, choose one of the points and substitute its coordinates along with the normal vector into the equation: ax + by + cz = d, where (a, b, c) is the normal vector and (x, y, z) are the coordinates of the point.
| 5. Is it possible to find the equation of a plane if you know the coordinates of a point on the plane and the normal vector of the plane? | |
Ans. Yes, it is possible to find the equation of a plane if you know the coordinates of a point on the plane and the normal vector of the plane. You can use the point-normal form of the equation: ax + by + cz = d, where (a, b, c) is the normal vector and (x, y, z) are the coordinates of the point. Simply substitute the values into the equation to find the constant term (d).
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