NDA Maths · Teaching notes

Properties of Triangle — NDA Maths

Properties of Triangle is the toughest yield in NDA Maths — 49 PYQs, 45% of them HARD. Everything connects the sides a, b, c to the angles A, B, C, and the marks come from choosing the right relation: the sine rule, the cosine rule, or a triangle identity that uses A + B + C = π. The notes teach in three movements, foundations first: (1) Sine & Cosine Rules — the side/angle notation, the sine rule (a/sin A = 2R), the cosine rule, the area formulas, how to read off the nature of a triangle, and the angle-ratio ↔ side-ratio links; (2) Triangle Identities — the consequences of A + B + C = π (sin(B+C) = sin A, half-angle forms), the tan A + tan B + tan C = tan A·tan B·tan C identity, and the cos 2A / sin² identities that detect a right angle; (3) In-circle & Regular Polygons — the inradius r = Δ/s, the circumradius R = abc/4Δ, the central-angle relation, and the inradius of a regular n-gon. Sine/cosine rules and the half/double-angle identities are the highest-leverage tools. Every PYQ is tagged.

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Subtopic notes

PYQ weightage by concept

13 concepts · 49 PYQs — where the marks actually sit, so you know what to drill first

Sine & Cosine Rules — Solving Triangles29 PYQs · 59%

Concept	PYQs	Share
The Sine Rule	7	14%
The Cosine Rule	7	14%
Determining the Nature of a Triangle	5	10%
Angle Ratios ↔ Side Ratios	5	10%
Sine Rule in Geometric Configurations	3	6%
Area of a Triangle	2	4%
Triangle Notation & Basic Relationsfoundation	—	—

Triangle Identities — A+B+C = π, Half & Double Angle14 PYQs · 29%

Concept	PYQs	Share
Half-Angle Formulas & Sum-to-Product	5	10%
Consequences of A + B + C = π	4	8%
The tan-Sum = tan-Product Identity	3	6%
cos 2A Sums & Right-Angle Detection	2	4%

In-circle, Circumcircle & Regular Polygons6 PYQs · 12%

Concept	PYQs	Share
Incircle, Circumcircle & the Central Angle	4	8%
Regular Polygon Geometry	2	4%

Formula & revision sheet

13 formulas · 3 gotchas across all subtopics — the exam-eve cheat-sheet

Sine & Cosine Rules — Solving Triangles

Formulas (7)

Triangle Notation & Basic Relations · Angle sum & semi-perimeter $A + B + C = \pi, \qquad s = \dfrac{a+b+c}{2}$
The Sine Rule · Sine rule $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R$
The Cosine Rule · Cosine rule $\cos C = \dfrac{a^2 + b^2 - c^2}{2ab}$
Determining the Nature of a Triangle · Right-angle tests $c^2 = a^2 + b^2 \iff C = 90^\circ \iff \cos^2 A + \cos^2 B + \cos^2 C = 1$
Area of a Triangle · Area formulas $\Delta = \tfrac12 ab\sin C = \sqrt{s(s-a)(s-b)(s-c)} = \dfrac{abc}{4R} = rs$
Angle Ratios ↔ Side Ratios · Sides proportional to sines $a : b : c = \sin A : \sin B : \sin C$
Sine Rule in Geometric Configurations · Sine rule in a sub-triangle $\dfrac{AB}{\sin(\angle ADB)} = \dfrac{AD}{\sin(\angle ABD)}$

Watch out for (2)

Side over sine of its OWN opposite angle
→ The Sine Rule
Put the opposite side as the one being squared on the left
→ The Cosine Rule

Triangle Identities — A+B+C = π, Half & Double Angle

Formulas (4)

Consequences of A + B + C = π · Half-angle complement $\sin\dfrac{B+C}{2} = \cos\dfrac{A}{2}, \qquad \sin(B+C) = \sin A$
The tan-Sum = tan-Product Identity · tan sum = tan product $\tan A + \tan B + \tan C = \tan A\,\tan B\,\tan C$
cos 2A Sums & Right-Angle Detection · Right-angle signature $\sin^2 A + \sin^2 B + \sin^2 C = 2 \iff \text{right-angled}$
Half-Angle Formulas & Sum-to-Product · Half-angle tangent $\tan\dfrac{A}{2} = \dfrac{r}{s-a} = \sqrt{\dfrac{(s-b)(s-c)}{s(s-a)}}$

Watch out for (1)

The sum equals the PRODUCT, only in a triangle
→ The tan-Sum = tan-Product Identity

In-circle, Circumcircle & Regular Polygons

Formulas (2)

Incircle, Circumcircle & the Central Angle · Inradius, circumradius, central angle $r = \dfrac{\Delta}{s}, \quad R = \dfrac{abc}{4\Delta}, \quad \angle BOC = 2\,\angle BAC$
Regular Polygon Geometry · Regular n-gon inradius $r = \dfrac{s}{2}\cot\dfrac{\pi}{n}, \qquad \text{interior angle} = \dfrac{(n-2)180^\circ}{n}$