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 \(\frac{a+b}{c}=\frac{\cos \left(\frac{A-B}{2}\right)}{\sin \frac{C}{2}}\)
\(\frac{b+c}{a}=\frac{\cos \left(\frac{B-C}{2}\right)}{\sin \frac{A}{2}}\)
\(\frac{c+a}{b}=\frac{\cos \left(\frac{C-A}{2}\right)}{\sin \frac{B}{2}}\)