# Inter 1st Year Maths 1A Properties of Triangles Formulas

Use these Inter 1st Year Maths 1A Formulas PDF Chapter 10 Properties of Triangles to solve questions creatively.

## Intermediate 1st Year Maths 1A Properties of Triangles Formulas

→ Sine Rule :
In ΔABC $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$ = 2R where R is the circumradius of ΔABC.

→ Cosine Rule :
a2 = b2 + c2 – 2bc. cos A ;
b2 = c2 + a2 – 2ca.cos B;
c2 = a2 + b2 – 2ab. cos C.

→ cos A = $$\frac{b^{2}+c^{2}-a^{2}}{2 b c}$$,
cos B = $$\frac{c^{2}+a^{2}-b^{2}}{2 c a}$$,
cos C = $$\frac{a^{2}+b^{2}-c^{2}}{2 a b}$$

→ a = b cos C + c cos B,b = c cos A + a cos C and c = a cos B + b cos A (Projection rule)

→ tan $$\frac{B-C}{2}=\frac{b-c}{b+c}$$ cot$$\frac{A}{2}$$ (Napier’s analogy or tangent rule)

• sin$$\frac{A}{2}$$ = $$\sqrt{\frac{(s-b)(s-c)}{b c}}$$
• cos$$\frac{A}{2}$$ = $$\sqrt{\frac{s(s-a)}{b c}}$$
• tan$$\frac{A}{2}$$ = $$\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}=\frac{\Delta}{s(s-a)}$$

→ Δ = area of ΔABC = $$\frac{1}{2}$$ bc sin A = $$\frac{1}{2}$$ ca sin B = $$\frac{1}{2}$$ ab sin C
= $$\sqrt{s(s-a)(s-b)(s-c)}=\frac{a b c}{4 R}$$
= 2R2 sin A sin B sin C

• r = $$\frac{\Delta}{s}$$
• r1 = $$\frac{\Delta}{s-a}$$
• r2 = $$\frac{\Delta}{s-b}$$
• r3 = $$\frac{\Delta}{s-c}$$

→ r = 4 R sin $$\frac{A}{2}$$ sin $$\frac{B}{2}$$ sin $$\frac{C}{2}$$; r1 = 4Rsin $$\frac{A}{2}$$ cos $$\frac{B}{2}$$ cos $$\frac{C}{2}$$

→ r = (s – a) tan $$\frac{A}{2}$$;
r1 = s tan $$\frac{A}{2}$$ = (s – c) cot $$\frac{B}{2}$$ = (s – b) cot $$\frac{C}{2}$$

→ Mollweide rule.
In ΔABC $$\frac{a+b}{c}=\frac{\cos \left(\frac{A-B}{2}\right)}{\sin \frac{C}{2}}$$
$$\frac{b+c}{a}=\frac{\cos \left(\frac{B-C}{2}\right)}{\sin \frac{A}{2}}$$
$$\frac{c+a}{b}=\frac{\cos \left(\frac{C-A}{2}\right)}{\sin \frac{B}{2}}$$