Question

Find the third vertex of the triangle, if two of its vertices are $(5,7)$ and $(0,3)$ and centroid at origin.
A. $(10,5)$
B. $(5,10)$
C. $( - 5, - 10)$
D. $( - 10, - 5)$

Answer 1

Hint: Given that the centroid is at the origin whose coordinates are given by $(0,0)$ .Use the centroid formula and substitute the given two vertices and place an imaginary variable in place of the third coordinate. Evaluate further by solving the equations to get the value of the third coordinates.

Formula used: The formula for centroid when three vertices of a triangle are given is,
Given that the vertices are denoted by $({x_1},{y_1}),({x_2},{y_2}),({x_3},{y_3})$ and the centroid is denoted by $({x_c},{y_c})$
${x_c} = \dfrac{{({x_1} + {x_2} + {x_3})}}{3}$ ; ${y_c} = \dfrac{{({y_1} + {y_2} + {y_3})}}{3}$

Complete step-by-step solution:
The given coordinates of the triangle are, $(5,7),(0,3)$
The denotation is as follows,
$({x_1},{y_1}) = (5,7)$ and
$({x_2},{y_2}) = (0,3)$
Also given that the centroid is at the origin which is, $(0,0)$
This is also denoted by $({x_c},{y_c}) = (0,0)$
To find the third coordinate we use the centroid formula which is,
Given that the vertices are denoted by $({x_1},{y_1}),({x_2},{y_2}),({x_3},{y_3})$ and the centroid is denoted by $({x_c},{y_c})$
$\Rightarrow {x_c} = \dfrac{{({x_1} + {x_2} + {x_3})}}{3}$ ; ${y_c} = \dfrac{{({y_1} + {y_2} + {y_3})}}{3}$
Firstly, let's solve for the $x$-coordinate of the third vertex.
Here, ${x_1} = 5;{x_2} = 0;{x_c} = 0$
On substituting we get,
$\Rightarrow 0 = \dfrac{{(5 + 0 + {x_3})}}{3}$
Multiply with $3$ on both sides of the equation.
$\Rightarrow 0 = 5 + {x_3}$
Now subtract $5$ on both sides of the equation.
We get,
$\Rightarrow - 5 = {x_3}$
Now, let’s solve for the $y$-coordinate of the third vertex.
Here, ${y_1} = 7;{y_2} = 3;{y_c} = 0$
On substituting we get,
$\Rightarrow 0 = \dfrac{{(7 + 3 + {y_3})}}{3}$
Multiply with $3$ on both sides of the equation.
$\Rightarrow 0 = 10 + {y_3}$
Now subtract $10\;$ on both sides of the equation.
We get,
$\Rightarrow - 10 = {y_3}$
$\therefore$ The third vertex for the given triangle is $( - 5, - 10)$

Option C is the correct answer.

Note: The coordinate of the centroid must not be confused with the coordinate of the vertex while calculating. The centroid of the triangle always lies inside the triangle or on the triangle. Whenever a vertex or a point is said to be at the origin, then the coordinates will be $(0,0)$.