Question

Find the area of triangle whose vertices are $\left( { - 5, - 1} \right)\left( {3, - 5} \right)\left( {5,2} \right)$.

Answer 1

Hint: In the given problem three vertices points of a triangle is given. To find the area the triangle we will apply the formula to find the area of triangle i.e. $\dfrac{1}{2}\left[ {\left( {{x_1}{y_2} + {x_2}{y_3} + {x_3}{y_1}} \right) - \left( {{x_2}{y_1} + {x_3}{y_2} + {x_1}{y_3}} \right)} \right]$ Points are given in the question. We will put the values in the given formula, thus we will get the correct answer.

Complete step by step solution: Formula: Area of triangle= $\dfrac{1}{2}\left[ {\left( {{x_1}{y_2} + {x_2}{y_3} + {x_3}{y_1}} \right) - \left( {{x_2}{y_1} + {x_3}{y_2} + {x_1}{y_3}} \right)} \right]$
Given that triangle whose vertices are $\left( { - 5, - 1} \right)\left( {3, - 5} \right)\left( {5,2} \right)$
From these points, we get the values
$
  {x_1} = - 5 \\
  {y_1} = - 1 \\
  {x_2} = 3 \\
  {y_2} = - 5 \\
  {x_3} = 5 \\
  {y_3} = 2 \\
 $
Put all these values in the above formula
The formula is:
Area of triangle= $\dfrac{1}{2}\left[ {\left( {{x_1}{y_2} + {x_2}{y_3} + {x_3}{y_1}} \right) - \left( {{x_2}{y_1} + {x_3}{y_2} + {x_1}{y_3}} \right)} \right]$
Now by putting all values
We get:
\[
  \dfrac{1}{2}\left[ {\left( { - 5 \times \left( { - 5} \right) + 3 \times 2 + 5 \times \left( { - 1} \right)} \right) - \left( {3 \times \left( { - 1} \right) + 5 \times \left( { - 5} \right) + \left( { - 5} \right) \times 2} \right)} \right] \\
   \Rightarrow \dfrac{1}{2}\left[ {\left( { - 25 + 6 - 5} \right) - \left( { - 3 - 25 - 10} \right)} \right] \\
   \Rightarrow \dfrac{1}{2}\left[ {\left( { - 30 + 6} \right) - \left( { - 3 - 25 - 10} \right)} \right] \\
   \Rightarrow \dfrac{1}{2}\left[ {\left( { - 24} \right) - \left( { - 38} \right)} \right] \\
   \Rightarrow \dfrac{1}{2}\left[ { - 24 + 38} \right] \\
   \Rightarrow \dfrac{1}{2}\left[ {14} \right] \\
   \Rightarrow 7{\text{SqUnits}} \\
 \]

Hence, we get the correct answer. The area of the triangle is 7 sq units.

Note: In this question, we have to remember the formula of area of the triangle, without this we cannot solve this problem, and then we have to put all the values in the formula from the question where vertices are given. Thus by calculating it we get the correct answer that is 7 sq units.