Question

The area of a triangle is 5 square units, two of its vertices are \[\left( {2,1} \right)\] and \[\left( {3, - 2} \right)\]. The third vertex lies on\[y = x + 3\]. The third vertex is
A) \[\left( {\dfrac{7}{2},\dfrac{3}{2}} \right)\]
B) \[\left( {\dfrac{{ - 3}}{2},\dfrac{3}{2}} \right)\]
C) \[\left( {\dfrac{{ - 3}}{2},\dfrac{{13}}{2}} \right)\]
D) \[\left( {\dfrac{7}{2},\dfrac{5}{2}} \right)\]

Answer 1

Hint:
Here in this question we will use the formula of the area of the triangle. Area of the triangle can be calculated using the coordinates of the vertex of the triangle. We will equate the formula of the area of the triangle to 5 to find the unknown terms. Further, we will solve the equation to get the coordinate of the third vertex.

Complete step by step solution:
We will assume the coordinate of the third vertex to be \[\left( {x,y} \right)\].
It is also given that the third vertex lies on \[y = x + 3\]………….. \[\left( 1 \right)\]
We will now write the formula of the area of the triangle in terms of the vertex of the triangle.
So, area of the triangle \[ = \dfrac{1}{2}\left[ {{x_1}({y_2} - {y_3}) + {x_2}({y_3} - {y_1}) + {x_3}({y_1} - {y_2})} \right]\]
Now, by equating this area of the triangle is equal to the 5 square units as given in the question. Therefore, we get
\[ \Rightarrow \dfrac{1}{2}\left[ {{x_1}({y_2} - {y_3}) + {x_2}({y_3} - {y_1}) + {x_3}({y_1} - {y_2})} \right] = 5\]
On cross multiplication, we get
\[ \Rightarrow \left[ {{x_1}({y_2} - {y_3}) + {x_2}({y_3} - {y_1}) + {x_3}({y_1} - {y_2})} \right] = 10\]
Now substituting the values of the coordinates in the above equation, we get
\[ \Rightarrow \left[ {2( - 2 - y) + 3(y - 1) + x(1 + 2)} \right] = 10\]
\[ \Rightarrow - 4 - 2y + 3y - 3 + 3x = \pm 10\]
By simplifying the above equation, we get
\[ \Rightarrow y + 3x = 17\]…………… (2)
\[ \Rightarrow y + 3x = - 3\]…………… (3)
So by solving the equation (1) and equation (2), we get
\[ \Rightarrow x + 3 + 3x = 17\]
Adding the like terms, we get
\[ \Rightarrow 4x = 14\]
Dividing both side by 4, we get
\[ \Rightarrow {\rm{x}} = \dfrac{7}{2}\]
And by solving the equation (1) and equation (3), we get
\[ \Rightarrow x + 3 + 3x = - 3\]
Adding the like terms, we get
\[ \Rightarrow 4x = - 6\]
Dividing both side by 4, we get
\[ \Rightarrow x = \dfrac{{ - 3}}{2}\]
Now by putting the value of \[x\] in equation (1) we will get the value of \[y\]. Therefore, we get
\[y = \dfrac{7}{2} + 3 = \dfrac{{13}}{2}\] for corresponding to the value of \[x = \dfrac{7}{2}\]
\[y = \dfrac{{ - 3}}{2} + 3 = \dfrac{3}{2}\] for corresponding to the value of \[x = \dfrac{{ - 3}}{2}\]
So, \[\left( {\dfrac{7}{2},\dfrac{{13}}{2}} \right)\] or\[\left( {\dfrac{{ - 3}}{2},\dfrac{3}{2}} \right)\] is the coordinates of the third vertex.

So, option B is correct.

Note:
Geometry is the branch of mathematics that deals with points, lines and shapes.
There are some basic definitions which we need to know:
A triangle is a polygon with three edges/sides and three vertices. Side is one of the straight line segments which is used to construct/draw a polygon.
When two or more lines cross each other in a plane, they are called intersecting lines and the point where these lines intersect is called a Point of Intersection or vertex.