Question

If the coordinates of the points A, B, and C be \[(4,4),(3, - 2),(3, - 16)\] respectively, then find the area of the triangle ABC.
A. 27
B. 15
C. 18
D. 7

Answer 1

Hints First write the formula of the area of a triangle, then substitute the given coordinates in the formula to obtain the required result.

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

Complete step by step solution
Substitute \[({x_1},{y_1}),({x_2},{y_2}),({x_3},{y_3})\] by \[(4,4),(3, - 2),(3, - 16)\] in the formula \[\dfrac{1}{2}\left[ {{x_1}{\rm{\;}}\left( {{y_{2 - }}{\rm{\;}}{y_3}{\rm{\;}}} \right) + {x_2}{\rm{\;}}\left( {{y_3} - {y_1}{\rm{\;}}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]\]and calculate to obtain the required area.
\[\dfrac{1}{2}\left[ {4\left( { - 2 + 16} \right) + 3( - 16 - 4) + 3(4 + 2)} \right]\]
\[ = \dfrac{1}{2}\left[ {56 - 60 + 18} \right]\]
\[ = \dfrac{{14}}{2}\]
=7

The correct option is “D”.

Note While calculating the area of the triangle when the Cartesian coordinates are given, one can also proceed by first plotting the triangle on an X-Y graph. This process can help in identifying the type of triangle that is whether it is an equilateral triangle, isosceles triangle, or right triangle. If we can identify that the triangle is one of them, we can easily calculate the area of the triangle, by using the respective formulas for these special types of triangles. This greatly reduces the time taken in calculating the area of the triangle. If the triangle is not of any special type then use just the general formula. This is also a good approach to doing this type of question.