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

Sine rule :
In ΔABC, \(\frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\sin \mathrm{B}}=\frac{\mathrm{c}}{\sin \mathrm{C}}\) = 2R Where R is the circum – radius.
⇒ a = 2R sin A, b = 2R sin B, c = 2R sin C
a : b : c = sin A : sin B : sin C.

Cosine rule :
In ΔABC,
a² = b² + c² – 2bc cos A
b² = c² + a² – 2ac cos B
c² = a² + b² – 2ab cos C
or
cos A = \(\frac{\mathrm{b}^{2}+\mathrm{c}^{2}-\mathrm{a}^{2}}{2 \mathrm{bc}}\)
cos B = \(\frac{c^{2}+a^{2}-b^{2}}{2 c a}\)
cos C = \(\frac{a^{2}+b^{2}-c^{2}}{2 a b}\) ⇒ cos A : cos B : cos C
= a(b² + c² – a²) : b(c² + a² – b²) : c(a² + b² – c²)

Projection rule :
In ΔABC

• a = b cos C + c cos B,
• b = c cos A + a cos C,
• c = s cos B + b cos A

Mollwiede’s rule :
In ΔABC

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

Similarly the other two can be written by symmetry.

Tangent rule (or) Napier’s analogy :
In ΔABC

Half angle formulae :

cot A = \(\frac{\mathrm{b}^{2}+\mathrm{c}^{2}-\mathrm{a}^{2}}{4 \Delta}\)
cot B = \(\frac{c^{2}+a^{2}-b^{2}}{4 \Delta}\)
cot C = \(\frac{a^{2}+b^{2}-c^{2}}{4 \Delta}\)

→ Area of ΔABC 1s given by

• Δ = \(\frac{1}{2}\)ab sin C = \(\frac{1}{2}\)bc sin A = \(\frac{1}{2}\) ca sin B
• Δ = \(\sqrt{s(s-a)(s-b)(s-c)}\).
• Δ = \(\frac{a b c}{4 R}\)
• Δ = 2R² sin A sin B sin C
• Δ = rs
• Δ = \(\sqrt{\mathrm{rr}_{1} \mathrm{r}_{2} \mathrm{r}_{3}}\)

→ If ‘r’ is radius of in circle and r₁, r₂, r₃ are the radii of ex-circles opposite to the vertices A, B, C of ΔABC respectively then
i. r = \(\frac{\Delta}{\mathrm{s}}\), r₁ = \(\frac{\Delta}{\mathrm{s-a}}\), r₂ = \(\frac{\Delta}{\mathrm{s-b}}\), r₃ = \(\frac{\Delta}{\mathrm{s-c}}\)

→ r = 4R sin\(\frac{\mathrm{A}}{2}\)sin\(\frac{\mathrm{B}}{2}\) sin\(\frac{\mathrm{C}}{2}\)

• r₁ = 4R sin\(\frac{\mathrm{A}}{2}\)cos\(\frac{\mathrm{B}}{2}\)cos\(\frac{\mathrm{C}}{2}\)
• r₂ = 4R cos\(\frac{\mathrm{A}}{2}\)sin\(\frac{\mathrm{B}}{2}\) cos\(\frac{\mathrm{C}}{2}\)
• r₃ = 4R cos\(\frac{\mathrm{A}}{2}\)cos\(\frac{\mathrm{B}}{2}\) sin\(\frac{\mathrm{C}}{2}\)

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

• r₁ = s tan\(\frac{A}{2}\) = (s – b)cot\(\frac{C}{2}\) = (s – c)cot\(\frac{B}{2}\)
• r₂ = s tan\(\frac{B}{2}\) = (s – c)cot\(\frac{A}{2}\) = (s – a)cot\(\frac{c}{2}\)
• r₁ = s tan\(\frac{A}{2}\) = (s – a)cot\(\frac{B}{2}\) = (s – b)cot\(\frac{A}{2}\)

→ \(\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}\)

→ rr₁r₂r₃ = Δ²

→ r₁r₂ + r₂r₃ + r₃r₁ = s²

→ r(r₁ + r₂ + r₃) = ab + bc + ca – s²

→ (r₁ – r)(r₂ + r₃) = a²

→ (r₂ – r)(r₃ + r₁) = b²

→ (r₃ – r)(r₁ + r₂) = c²

→ a = (r₂ + r₃)\(\sqrt{\frac{r r_{1}}{r_{2} r_{3}}}\)

→ b = (r₃ + r₁)\(\sqrt{\frac{r r_{2}}{r_{3} r_{1}}}\)

→ c = (r₁ + r₂)\(\sqrt{\frac{r r_{3}}{r_{1} r_{2}}}\)

→ r₁ – r = 4Rsin²\(\frac{A}{2}\)

→ r₂ – r = 4Rsin²\(\frac{B}{2}\)

→ r₃ – r = 4Rsin²\(\frac{C}{2}\)

→ r₁ + r₂ = 4R cos²\(\frac{C}{2}\)

→ r₂ + r₃ = 4R cos²\(\frac{A}{2}\)

→ r₃ + r₁ = 4R cos²\(\frac{B}{2}\)

→ r₁ + r₂ + r₃ = 4R

→ r + r₂ + r₃ – r₁ = 4R cos A

→ r + r₁ + r₃ – r₂ = 4R cos B

→ r + r₁ + r₂ – r₃ = 4R cos C

→ In an equilateral triangle of side ‘a’
area = \(\frac{\sqrt{3} a^{2}}{4}\)
R = a/√3
r = R/2
r₁ = r₂ + r₃ = 3R/2
r : R : r₁ = 1 : 2 : 3

In circle:
The circle that touches the three sides of a triangle ABC internally is called the “in circle” or inscribed” of its triangle. The centre of the circle is called Incentre denoted by I the radius of the circle is denoted by inradius denoted by Y

→ In a triangle ABC

Excircle:
The circle that touches the side BC (opposite to angle A) internally and the other two sides AB and AC externally is called Excircle. The centre of this circle is called excentre opposite to ‘A’. denoted by I₁. The radius of this circle is called ex-radius, denoted by r₁

|||^ly exradius opposite to angle B is denoted by r₂. The centre of this excircle is denoted by I₂ exradius opposite to angle C is denoted by r₃. The centre of this ex-circle is denoted by I₃

In a triangle ABC

• r₁ = \(\frac{\Delta}{s-a}\)
• r₂ = \(\frac{\Delta}{s-b}\)
• r₃ = \(\frac{\Delta}{s-c}\)

→ r₁ = s tan A/2

→ r₂ = s tan B/2

→ r₂ = s tan C/2

→ r₁ = (s – c)cot\(\frac{B}{2}\)
= (s- b)cot\(\frac{C}{2}\)

→ r₁ = (s – a)cot\(\frac{C}{2}\)
= (s – c)cot\(\frac{A}{2}\)

→ r₁ = (s – a)cot\(\frac{B}{2}\)
= (s – c)cot\(\frac{A}{2}\)

→ r₁ = \(\frac{a}{\tan \frac{B}{2}+\tan \frac{C}{2}}\)

→ r₂ = \(\frac{b}{\tan \frac{C}{2}+\tan \frac{A}{2}}\)

→ r₃ = \(\frac{c}{\tan \frac{A}{2}+\tan \frac{B}{2}}\)

→ r₁ = 4Rsin\(\frac{A}{2}\)cos\(\frac{B}{2}\)cos\(\frac{C}{2}\)

→ r₂ = 4Rcos\(\frac{A}{2}\)sin\(\frac{B}{2}\)cos\(\frac{C}{2}\)

→ r₃ = 4Rcos\(\frac{A}{2}\)cos\(\frac{B}{2}\)sin\(\frac{C}{2}\)