Properties and Solution of Triangle Notes for Class 12 & IIT JEE

The most versatile formula is Δ = ½ ab sinC, used when two sides and the included angle are known. When all three sides are given, use Heron's Formula: Δ = √s(s−a)(s−b)(s−c). For problems involving R or r, use Δ = abc/4R or Δ = rs respectively.

The circumradius (R) is the radius of the circle passing through all three vertices, given by R = abc/4Δ. The inradius (r) is the radius of the circle inscribed inside the triangle, given by r = Δ/s. In any triangle, R ≥ 2r, with equality only for an equilateral triangle.

The Sine Rule states that a/sinA = b/sinB = c/sinC = 2R. Use it when at least one side and its opposite angle are known (AAS or ASA case), or when the problem involves the circumradius R. It appears in nearly every JEE Main paper and is the starting point for most triangle problems

Heron's Formula gives the area of a triangle purely from its three sides: Δ = √s(s−a)(s−b)(s−c), where s = (a+b+c)/2 is the semi-perimeter. It is derived by substituting the Cosine Rule expression for cosA into the area formula Δ = ½ bc sinA and simplifying using the identity sin²A + cos²A = 1.

Yes. The CBSE Class 12 syllabus includes properties of triangles under Trigonometry. Board exams typically test the Sine Rule, Cosine Rule, and area formulas at a straightforward level. Mastering JEE-level problems on this chapter ensures full marks in boards with no extra effort.

Ex-radii (r₁, r₂, r₃) are the radii of three excircles, each tangent to one side and the extensions of the other two sides. They are defined as r₁ = Δ/(s−a). JEE Advanced tests identities like r₁ + r₂ + r₃ − r = 4R and products involving ex-radii, making them a high-value topic to master.