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

The direction ratios of OA are Area of a Triangle and Equation of a Plane | Mathematics (Maths) Class 12 - JEE

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

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

∴ the d.c.’ s of OA are Area of a Triangle and Equation of a Plane | Mathematics (Maths) Class 12 - JEE

and the d.c.’s of OB are Area of a Triangle and Equation of a Plane | Mathematics (Maths) Class 12 - JEE

 

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

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


Hence the area of ΔOAB

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


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

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


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

∴ the required area Δ = Area of a Triangle and Equation of a Plane | Mathematics (Maths) Class 12 - JEE

Ex.7 A plane is passing through a point P(a, –2a, 2a), Area of a Triangle and Equation of a Plane | Mathematics (Maths) Class 12 - JEE 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 Area of a Triangle and Equation of a Plane | Mathematics (Maths) Class 12 - JEE

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.

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

The document Area of a Triangle and Equation of a Plane | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Area of a Triangle and Equation of a Plane - Mathematics (Maths) Class 12 - JEE

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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