## Straight Lines

# 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 a
_{1}x + b_{1}y + c_{1}= 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 P
_{1}, P_{2}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 a
_{1}x + b_{1}y + c_{1}= 0 , a_{2}x + b_{2}y + c_{2}= 0 , a_{2}x + b_{2}y + d_{1}= 0 and a_{2}x + b_{2}y + d_{2}= 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 ‘d
_{1}’ be the distance between the parallel lines a_{1}x + b_{1}y + k_{1}= 0, a_{1}x + b_{1}y + k_{2}= 0 then the figure formed by four lines is

i) A square if d_{1}= d_{2}and aa_{1}+ bb_{1}= 0

ii) Rhombus if d_{1}= d_{2}and aa_{1}+ bb_{1}≠ 0

iii) Rectangle if d_{1}≠ d_{2}and aa_{1}+ bb_{1}= 0

iv) Parallelogram if d_{1}≠ d_{2}and aa_{1}+ bb_{1}≠ 0 - The equations of bisectors of angle between the lines a
_{1}x + b_{1}y + c_{1}= 0 and a_{2}x + b_{2}y + c_{2}= 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 (x
_{1}y_{1}) (x_{2}y_{2}) (x_{3}y_{3}) 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 = m_{1}x + c_{1}, y = m_{2}x + c_{2} and y = m_{3}x + c_{3} 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 a_{1}x + b_{1}y + c_{1} = 0, a_{2}x + b_{2}y + c_{2} = 0, a_{1}x + b_{1}y + d_{1} = 0 and a_{2}x + b_{2}y + d_{2} = 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 (x_{1} , y_{1}) and (x_{2}, y_{2}) 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 A_{1} (x_{1} , y_{1}), A_{2} (x_{2} , y_{2}),...., A_{n}(x_{n} , y_{n}) \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]