Question

Find the area of the triangle whose vertices are \[\left( {1,2} \right),\left( { - 3,4} \right)\] and \[\left( { - 5, - 6} \right)\].

Answer 1

Hint: When coordinates are given in the question and when we have to find the area of the triangle we use the formula, \[\dfrac{1}{2}\left| {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right|\] where \[\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)\] and \[\left( {{x_3},{y_3}} \right)\] are the vertices of the triangle, and this is derived by finding the length of the base of the triangle and its height using the distance formula.

Complete step-by-step answer:
Given the vertices of the triangle are \[\left( {1,2} \right),\left( { - 3,4} \right)\] and \[\left( { - 5, - 6} \right)\].
This is represented in the given diagram,

Let the vertices of the triangle are represented as,
$\Rightarrow$ \[\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)\]and\[\left( {{x_3},{y_3}} \right)\],
And the area of the triangle can be determined by using the formula,
$\Rightarrow$ \[\dfrac{1}{2}\left| {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right|\],
Now, now substituting the values in the formula we get,
Area of triangle\[ = \dfrac{1}{2}\left| {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right|\],
Now here, \[{x_1} = 1,{y_1} = 2,{x_2} = - 3,{y_2} = 4,{x_3} = - 5,{y_3} - 6\],
Then the formula will be,
$\Rightarrow$ Area of triangle\[ = \dfrac{1}{2}\left| {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right|\]
$\Rightarrow$ Area of triangle\[ = \dfrac{1}{2}\left| {1\left( {2 - ( - 6)} \right) + ( - 3)\left( { - 6 - 2} \right) + ( - 5)\left( {2 - 4} \right)} \right|\],
Now simplifying inside the brackets we get,
$\Rightarrow$ Area of triangle\[ = \dfrac{1}{2}\left| {1\left( {2 + 6} \right) + ( - 3)\left( { - 8} \right) + ( - 5)\left( { - 2} \right)} \right|\],
$\Rightarrow$ Area of triangle\[ = \dfrac{1}{2}\left| {1\left( 8 \right) + ( - 3)\left( { - 8} \right) + ( - 5)\left( { - 2} \right)} \right|\],
Now multiplying inside the absolute value we get,
Area of triangle\[ = \dfrac{1}{2}\left| {8 + 24 - 10} \right|\],
Now adding inside the absolute value get,
$\Rightarrow$ Area of triangle\[ = \dfrac{1}{2}\left| {32 - 10} \right|\]
$\Rightarrow$ Area of triangle\[ = \dfrac{1}{2}\left| {22} \right|\],
$\Rightarrow$ Area of triangle\[ = 11\],
Therefore the area of triangle whose vertices are is \[\left( {1,2} \right),\left( { - 3,4} \right)\]and\[\left( { - 5, - 6} \right)\]\[ = 11\]sq. units.

\[\therefore \] The area of triangle whose vertices are is \[\left( {1,2} \right),\left( { - 3,4} \right)\]and\[\left( { - 5, - 6} \right)\]\[ = 11\]sq. Units.

Note:
In these types of questions students may get confused and make mistakes in substituting the values in the area formula as there are three points they should note the vertices in the formula and also there is a chance that the calculations mistakes can be done, so students should do calculations carefully.