Answer:
b. [tex](\frac{3}{2},-2)[/tex]
Step-by-step explanation:
Given:
A triangle with coordinates A(-1, 0), B(-1, -4), and C(4, -4).
As clear from the graph the two legs of the triangle AB and BC are perpendicular to each other as BC is parallel to x-axis and AB is parallel to y-axis.
Therefore, the given triangle is a right angled triangle with right angle at B and AC as the hypotenuse.
Circumcenter is the center of a circle that passes through the three vertices of the given triangle.
Now, we know that, circumcenter of a right angled triangle always lies on the midpoint of its hypotenuse. Therefore, the circumcenter is the midpoint of line segment AC.
By midpoint formula, the midpoint of two points [tex](x_1,y_1)\ and\ (x_2,y_2)[/tex] is given as:
Midpoint = [tex](\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]
Therefore, the circumcenter for the points A(-1, 0) and C(4, -4) is given as:
Circumcenter = [tex](\frac{-1+4}{2},\frac{0-4}{2})=(\frac{3}{2},\frac{-4}{2})=(\frac{3}{2},-2)[/tex]
Therefore, the circumcenter of the given triangle is at the point [tex](\frac{3}{2},-2)[/tex].
So, option (b) is correct.
