# 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 = mi x + ci , 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

### View the Topic in this video From 00:40 To 51:10

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1. The area of the triangle formed by the lines y = m1x + c1, y = m2x + c2 and y = m3x + c3 is \tt \triangle = \frac{1}{2}\begin{vmatrix}\sum \frac{\left(c_{1}-c_{2}\right)^{2}}{m_{1}-m_{2}} \end{vmatrix}.
2. Area of the triangle formed by the line ax + by + c = 0 with the coordinate axes is \tt \triangle = \frac{c^{2}}{2\mid ab \mid}.
3. Area of rhombus formed by ax ± by ± c = 0 is \tt \begin{vmatrix}\frac{2c^{2}}{ab} \end{vmatrix}.
4. Area of the parallelogram formed by the lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0, a1x + b1y + d1 = 0 and a2x + b2y + d2 = 0 is \tt \begin{vmatrix}\frac{\left(d_{1}-c_{1}\right)\left(d_{2}-c_{2}\right)}{a_{1}b_{2}-a_{2}b_{1}} \end{vmatrix}.
5. Given two vertices (x1 , y1) and (x2, y2) of an equilateral ΔABC, then its third vertex is given by. \tt \left[\frac{x_{1}+x_{2}\pm \sqrt{3}\left(y_{1}-y_{2}\right)}{2},\frac{y_{1}+y_{2}\mp \sqrt{3}\left(x_{1}-x_{2}\right)}{2}\right]
6. Area of a polygon of n-sides with vertices A1 (x1 , y1), A2 (x2 , y2),...., An(xn , yn) \tt =\frac{1}{2}\left[\begin{vmatrix}x_{1} & y_{1} \\ x_{2} & y_{2} \end{vmatrix}+\begin{vmatrix}x_{2} & y_{2} \\ x_{3} & y_{3} \end{vmatrix}+...+\begin{vmatrix}x_{n} & y_{n} \\ x_{1} & y_{1} \end{vmatrix}\right]