Question

If \[\left( { - 4,0} \right)\] and \[\left( {1, - 1} \right)\] are two vertices of a triangle of area 4 square units, then its third vertex lies on
\[y = x\]
\[5x + y + 12 = 0\]
\[x + 5y - 4 = 0\]
\[x + 5y + 12 = 0\]

Answer 1

Hint:
Here, we will use the concept of vertices and coordinate geometry to find the equation of the third vertex of the triangle. We will substitute the given vertices into the formula of the area of the triangle. We will simplify it further to get the coordinates of the third vertex.

Formula used:
Area of the triangle = \[\dfrac{1}{2}\left( {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right)\] where \[\left( {{x_1},{y_1}} \right)\] , \[\left( {{x_2},{y_2}} \right)\] , \[\left( {{x_3},{y_3}} \right)\] are the coordinates of the triangle.

Complete step by step solution:
We will first draw the triangle using the given information.

Let \[(x,y)\] be the third vertex of the triangle.
Area of the triangle = \[\dfrac{1}{2}\left( {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right)\]
Substituting the vertices \[\left( { - 4,0} \right)\], \[\left( {1, - 1} \right)\] and area as 4 in the above formula, we get
\[ \Rightarrow 4 = \dfrac{1}{2}\left( { - 4( - 1 - y) + 1(y - 0) + x(0 + 1)} \right)\]
Multiplying by 2 on both the sides, we get
\[ \Rightarrow 8 = - 4( - 1 - y) + 1(y - 0) + x(1)\]
Multiplying by terms on the R.H.S, we get
\[ \Rightarrow 8 = 4 + 4y + y + x\]
Adding the like terms, we get
\[ \Rightarrow 8 = 4 + 5y + x\]
Subtracting like terms, we get
\[\begin{array}{l} \Rightarrow x + 5y + 4 - 8 = 0\\ \Rightarrow x + 5y - 4 = 0\end{array}\]
Therefore, the third vertex of the triangle lies on \[x + 5y - 4 = 0\].

Hence, option C is correct.

Note:
By using the vertex of the triangle we can find the coordinates of the third vertex if either the \[x\] co-ordinate or \[y\] co-ordinate is known. We should be clear about the area of the triangle formula since there are too many formulae for the area of the triangle. A vertex of a geometrical shape is a point where two sides of the shape meet. Vertices usually define a geometric shape, when the same shape is defined inside the coordinate plane.