Question

The orthocenter of a triangle ABC is B and the circumcenter is S(a,b). If A is the origin then the coordinates of C are
A) \[\left( {2a,2b} \right)\]
B) \[\left( {\dfrac{a}{2},\dfrac{b}{2}} \right)\]
C) \[\left( {\sqrt {{a^2} + {b^2}} ,0} \right)\]
D) None

Answer 1

Hint:
Since B is the orthocenter, ABC is a right-angled triangle, right angled at B which makes AC the hypotenuse of the triangle. This means that the circumcenter S is going to lie on AC, because it is a known fact that the circumcenter of a right triangle lies on its hypotenuse and is exactly found at the center of the hypotenuse.

Formula used:
We shall be using the midpoint formula for calculating the coordinates of the vertex C, which is:
The midpoint of the points \[\left( {{x_1},{y_1}} \right)\]and \[\left( {{x_2},{y_2}} \right)\] is –
\[\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)\] …(i)

Complete step by step solution:
We have to calculate the coordinates of vertex C. Let the coordinates of C be \[\left( {x,y} \right)\].

It has been given that the coordinates of vertex A is \[\left( {0,0} \right)\]because it is written in the question that A is at the origin.
B is given to be the orthocenter of the triangle ABC; hence, it means that the AC is hypotenuse of the triangle, making the triangle ABC to be right-angled at B.
Now, since AC is the hypotenuse of the triangle, the circumcenter of the triangle is found at the center of the hypotenuse, because it is a known fact that the circumcenter of a right triangle lies on its hypotenuse and is exactly found at the center of the hypotenuse.
Thus, from the above statements we can deduce the fact that we can find the required coordinates using the midpoint formula.
Here, S is the midpoint of A and C.

So, if we put in the coordinates of A \[\left( {0,0} \right)\] and C \[\left( {x,y} \right)\] into the midpoint formula, we get the coordinates of S \[\left( {a,b} \right)\], hence:
\[\left( {\dfrac{{x + 0}}{2},\dfrac{{y + 0}}{2}} \right) = \left( {a,b} \right)\]
\[\left( {\dfrac{x}{2},\dfrac{y}{2}} \right) = \left( {a,b} \right)\]
Now, equating the ordinate and abscissa of the 2 ordered pairs, we get:
\[\dfrac{x}{2} = a\] and \[\dfrac{y}{2} = b\]
or, \[x = 2a\] and \[y = 2b\]
Hence, the coordinates of C are \[\left( {2a,2b} \right).\]

Thus, the correct answer is the option A \[\left( {2a,2b} \right)\]

Note:
For solving such questions, first determine what we have to evaluate, what is given and note them down. Then, write down any relation, formula which correlates the given and the to-be-found. And apply it to mold the given equation in the favor of the parameters and evaluate the result.