Question

The coordinates of the vertices of a triangle are (1, 2), (3, 4) and (4, 6) :

Answer 1

(a) Centroid: $(\frac{8}{3}, 4)$

(b) Circumcenter: $(-\frac{7}{2}, \frac{17}{2})$, Circumradius: $\frac{5\sqrt{10}}{2}$

(c) Orthocenter: $(15, -5)$

Answer 2

1. Centroid: The centroid of a triangle is the average of the coordinates of its vertices. For vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, the centroid is $(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})$.
2. Circumcenter: The circumcenter is the point equidistant from all three vertices. It is found by solving the system of equations formed by setting the squared distances from the circumcenter to each pair of vertices equal. For a right-angled triangle, the circumcenter is the midpoint of the hypotenuse.
3. Circumradius: The circumradius is the distance from the circumcenter to any of the vertices.
4. Orthocenter: The orthocenter is the intersection point of the altitudes of the triangle. An altitude is a line segment from a vertex perpendicular to the opposite side. For a right-angled triangle, the orthocenter is the vertex where the right angle is located.