Properties of Triangle — NDA Maths

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 (8)

• Side over sine of its OWN opposite angle
  → The Sine Rule
• The ratio is $2R$, not $R$
  → The Sine Rule
• The ambiguous SSA case can give TWO triangles
  → The Sine Rule
• Put the opposite side as the one being squared on the left
  → The Cosine Rule
• A negative cosine means an OBTUSE angle
  → The Cosine Rule
• Test $a^2+b^2 = c^2$ on the LARGEST side only
  → Determining the Nature of a Triangle
• Area uses $\sin$ of the angle, not $\cos$
  → Area of a Triangle
• Heron's $s$ is the SEMI-perimeter
  → Area of a Triangle

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 (2)

• $\sin(B+C) = +\sin A$, but $\cos(B+C) = -\cos A$
  → Consequences of A + B + C = π
• The sum equals the PRODUCT, only in a triangle
  → The tan-Sum = tan-Product Identity

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
  $$r = \dfrac{s}{2}\cot\dfrac{\pi}{n}, \qquad \text{interior angle} = \dfrac{(n-2)180^\circ}{n}$$

Watch out for (2)

• Inradius $r$ vs circumradius $R$ — different formulas
  → Incircle, Circumcircle & the Central Angle
• Central angle is TWICE the inscribed angle
  → Incircle, Circumcircle & the Central Angle