Problems Related to Triangle and Quadrilaterals

Tips :

• The area of triangle formed by the line \tt \frac{x}{a}+ \frac{y}{b}=1 with the co-ordinate axis is 1/2|ab|
• The area of triangle formed by line ax + by + c = 0 with the co-ordinate axis is \tt \frac{c^{2}}{2\mid ab \mid}
• Area of the rhombus a|x|+b|y|+c= 0 is \tt \frac{2c^{2}}{\mid ab \mid}
• The area of triangle formed by lines a₁ x + b₁ y + c₁ = 0 , I = 1, 2, 3 is \tt \frac{1}{2}\begin{vmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3} \end{vmatrix}
• The area of triangle formed by lines y = m_i x + c_i , i = 1, 2, 3 is \tt \frac{1}{2}\begin{vmatrix}\sum \frac{\left(c_{1}-c_{2}\right)^{2}}{m_{1}-m_{2}} \end{vmatrix}
• If P₁ , P₂ are distance between parallel sides and ‘θ’ is angle between adjacent sides of parallelogram then its area is \tt \begin{vmatrix}\frac{P_{1}P_{2}}{sin \theta} \end{vmatrix}
• Area of parallelgram whose sides are a₁ x + b₁ y + c₁ = 0 , a₂ x + b₂ y + c₂ = 0 , a₂ x + b₂ y + d₁ = 0 and a₂ x + b₂ y + d₂ = 0 is \tt \begin{vmatrix}\frac{\left(c_{1}-c_{2}\right)\left(d_{1}-d_{2}\right)}{a_{1}b_{2}-a_{2}b_{1}} \end{vmatrix}

Tricks :

• Let ‘d₁’ be the distance between the parallel lines a₁ x + b₁ y + k₁ = 0, a₁ x + b₁ y + k₂ = 0 then the figure formed by four lines is
  i) A square if d₁ = d₂ and aa₁ + bb₁ = 0
  ii) Rhombus if d₁ = d₂ and aa₁ + bb₁ ≠ 0
  iii) Rectangle if d₁ ≠ d₂ and aa₁ + bb₁ = 0
  iv) Parallelogram if d₁ ≠ d₂ and aa₁ + bb₁ ≠ 0
• The equations of bisectors of angle between the lines a₁ x + b₁ y + c₁ = 0 and a₂ x + b₂ y + c₂ = 0 are \tt \frac{a_{1}x+b_{1}y+c_{1}}{\sqrt{a_1^2+b_1^2}}=\pm \frac{a_{2}x+b_{2}y+c_{2}}{\sqrt{a_2^2+b_2^2}}
• Ortho centre of ΔABC is when (x₁ y₁) (x₂ y₂) (x₃ y₃) are parts \tt \left(\frac{x_{1}\tan A +X_{2}\tan B+x_{3}\tan C}{\tan A + \tan B+\tan C}, \frac{y_{1}\tan A+y_{2}\tan B+y_{3}\tan C}{\tan A + \tan B+\tan C}\right)
• Circum centre of ΔABC is \tt \left(\frac{x_{1}\sin 2A +X_{2}\sin 2 B+x_{3}\sin 2C}{\sin 2A + \sin 2B+\sin 2C}, \frac{y_{1}\sin 2A +y_{2}\sin 2 B+y_{3}\sin 2C}{\sin 2A + \sin 2B+\sin 2C}\right)
• If a transversal cuts the sides BC, CA, AB of a triangle in D, E, F respectively then \tt \frac{BD}{DC}\times \frac{CE}{EA}\times \frac{AF}{FB}=-1