Question

The equations of the sides of a triangle are $x + y - 5 = 0,x - y + 1 = 0$ and $y - 1 = 0$ . Then find the coordinate of the circumcentre.
A. $(2,1)$
B. $(1,2)$
C. $(2, - 2)$
D. $(1, - 2)$

Answer 1

Hint: First obtain the slope of the first two lines and conclude that they are perpendicular to each other. Then substitute 1 for y in first two lines and obtain the values of x to obtain the coordinates. Then sum the x coordinates and the y coordinates and divide them by 2 to obtain the required result.

Formula Used:
The slope intercept form of a line is $y = mx + b$ , where m is the slope and b is the y intercept of the line.
Two lines with slope a and b are perpendicular to each other if$a.b = - 1$ .
The circumcentre of a right-angle triangle is the middle point of its hypotenuse.
The middle points of a line with end vertices $(p,q),(r,s)$ is $\left( {\dfrac{{p + r}}{2},\dfrac{{q + s}}{2}} \right)$.

Complete step by step solution:
Given first line is $x + y - 5 = 0$
The line can be written as,
$y = 5 - x$
Hence, the slope is $a = - 1$ .
The equation of the second line is $x - y + 1 = 0$ .
This line can be written as $y = x + 1$ .
Here, the slope is $b = 1$
Therefore, $a.b = - 1$
Hence, the lines $x + y - 5 = 0$and $x - y + 1 = 0$are perpendicular to each other.
Now, from the third line we have, $y = 1$ .
Substitute 1 for y in the first line we have,
$x + 1 - 5 = 0$
$x = 4$
Substitute 1 for y in the second line we have,
$x - 1 + 1 = 0$
$x = 0$
Now, obtain the middle point of the points $(4,1),(0,1)$ .
So, the mid-points is $\left( {\dfrac{{4 + 0}}{2},\dfrac{{1 + 1}}{2}} \right)$,
That is $(2,1)$

Option ‘A’ is correct

Note: Frequently fail to identify that the given two lines are perpendicular to each other and calculate to obtain the required answer; instead, identify that the lines are perpendicular and use the fact that the circumcentre of a right-angle triangle is the middle point of its hypotenuse and calculate to obtain the answer.