Question

In triangle ABC vertices are $A\left( {3,41} \right)$, $B\left( {{X_2},42} \right)$ and $C\left( {{X_3},43} \right)$, then find the area.

Answer 1

- Hint: Generally in these type of question or we say for finding the area of triangle whose vertices is given we can use the formula i.e.
For vertices $A\left( {3,41} \right)$, $B\left( {{X_2},42} \right)$ and $C\left( {{X_3},43} \right)$, the area of triangle ABC is given by:
$\dfrac{1}{2}\left[ {{X_1}\left( {{Y_2} - {Y_3}} \right) + {X_2}\left( {{Y_3} - {Y_1}} \right) + {X_3}\left( {{Y_1} - {Y_2}} \right)} \right]$
If you cannot remember this formula then you can generate this formula by using determinant. For example: area of triangle with vertices $A\left( {{X_1},{Y_1}} \right),B\left( {{X_2},{Y_2}} \right),C\left( {{X_3},{Y_3}} \right)$ is
                           $\dfrac{1}{2}\left( {\begin{array}{*{20}{c}}
  {{X_1}}&{{Y_1}}&1 \\
  {{X_2}}&{{Y_2}}&1 \\
  {{X_3}}&{{Y_3}}&1
\end{array}} \right)$
We can easily find the area of any triangle whose vertices will be given by using this formula.

Complete step-by-step solution:
Let $\vartriangle ABC$ is

Vertices of triangle ABC is $A\left( {3,41} \right)$, $B\left( {{X_2},42} \right)$and $C\left( {{X_3},43} \right)$
Now, area of $\vartriangle ABC$= $\dfrac{1}{2}\left[ {{X_1}\left( {{Y_2} - {Y_3}} \right) + {X_2}\left( {{Y_3} - {Y_1}} \right) + {X_3}\left( {{Y_1} - {Y_2}} \right)} \right]$
Here,
${X_1} = 3$, ${Y_1} = 41$
${X_2} = {X_2}$, ${Y_2} = 42$
${X_3} = {X_3}$, ${Y_3} = 43$
Putting values,
Area of $\vartriangle ABC$= $\dfrac{1}{2}\left[ {3\left( {42 - 43} \right) + {X_2}\left( {43 - 41} \right) + {X_3}\left( {41 - 42} \right)} \right]$
$ = \dfrac{1}{2}\left[ {3\left( { - 1} \right) + {X_2}\left( 2 \right) + {X_3}\left( { - 1} \right)} \right]$
$ = \dfrac{1}{2}\left( { - 3 + 2{X_2} - {X_3}} \right)$ Square units.
$ = \dfrac{{ - 1}}{2}\left( {3 - 2{X_2} + {X_3}} \right)$Square units.

Note:
Make the calculation carefully so that you should not make any mistake. One can make a mistake while putting values of points in the formula used. So be careful while putting values. In any triangle if vertices are given of that triangle then we can find the area of that triangle easily by using this formula.