Properties and Solution of Triangle Notes for Class 12 & IIT JEE
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eSaral provides you with complete study material to prepare for IIT JEE, NEET and Boards Examination. Here are the detailed Properties and Solution of Triangle with Important Questions that will also help in IIT JEE and boards preparation.
Frequently Asked Questions
Find answers to common questions.
How do I find the area of a triangle in JEE problems?
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.
What is the difference between inradius and circumradius?
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.
What is the Sine Rule and when should I use it in JEE?
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
What is Heron's Formula and how is it derived?
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.
Is Solution of Triangle important for Class 12 Boards as well?
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.
What are ex-radii and why are they important for JEE Advanced?
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.