Question

If \[A({x_1},{y_1}),B({x_2},{y_2}),C({x_3},{y_3})\] are the vertices of a triangle, then what is the excentre with respect to B?
A.\[\left( {\dfrac{{a{x_1} - b{x_2} + c{x_3}}}{{a - b + c}},\dfrac{{a{y_1} - b{y_2} + c{y_3}}}{{a - b + c}}} \right)\]
B. \[\left( {\dfrac{{a{x_1} + b{x_2} - c{x_3}}}{{a + b - c}},\dfrac{{a{y_1} + b{y_2} - c{y_3}}}{{a + b - c}}} \right)\]
C. \[\left( {\dfrac{{a{x_1} - b{x_2} - c{x_3}}}{{a - b - c}},\dfrac{{a{y_1} - b{y_2} - c{y_3}}}{{a - b - c}}} \right)\]
D. None of these

Answer 1

Hints Write the formula of the excentre of a triangle, then substitute the given vertices in the formula to obtain the required answer.

Complete step by step solution
The formula of excentre of a triangle PQR with respect to Q is \[\left( {\dfrac{{a{p_1} - b{p_2} + c{p_3}}}{{a - b + c}},\dfrac{{a{q_1} - b{q_2} + c{q_3}}}{{a - b + c}}} \right)\], where \[P({p_1},{q_1}),Q({p_2},{q_2}),R({p_3},{q_3})\]are the vertices of the triangle and a, b, c are the distance between BC, CA and Ab respectively.

The required formula is \[\left( {\dfrac{{a{x_1} - b{x_2} + c{x_3}}}{{a - b + c}},\dfrac{{a{y_1} - b{y_2} + c{y_3}}}{{a - b + c}}} \right)\].

The correct option is “A”

Additional information The angle bisector of a triangle is the median of the triangle. The median is the line that joins the middle point of every vertex with the opposite vertex of the triangle. Three medians of the triangle divide the triangle into six equal parts. The point at which the bisector of one interior angle and two bisectors of exterior angles intersect is called the excentre.

Note Sometimes students calculate the whole formula and then write but for this question that is not needed you just have to identify the correct answer.