Geometry Triangles – Circumcentre, Orthocentre

This GRE^® quant practice question tests your understanding of properties of triangles in Geometry. Concepts tested include properties of right triangles, orthocenter and circumcenter of triangles. Also, basic properties of coordinate geometry including coordinates of mid point of a line segment, equation of a line, computing intercepts of a line and distance between two points are tested in this question.

Consider the diagram below where the line 4x + 3y = 24 forms a triangle with the coordinate axes.

Thus, orthocenter for this triangle lies at (0, 0).

The circumcenter of any triangle is the point of intersection of the 3 perpendicular bisectors of the triangle. The circumcenter is the center of the circle that circumscribes the triangle.

Thus, mid-point of the segment joined by the points (6, 0) and (0, 8) is the circumcenter of this triangle.

The coordinates of the mid point of (6, 0) and (0, 8)  = (, ) = (3, 4)

Thus, the distance between (0, 0) and (3, 4) is the distance between the orthocenter and circumcenter of this triangle.

Distance between (0, 0) and (3, 4) = = 5 units.

The line joining the orthocenter and the circumcenter is nothing but the median to the hypotenuse.

We know that the median to the hypotenuse is the circumradius.

The distance between the center of a circle to any point on its circumference will measure the radius of the circle. The 3 vertices of the triangle lie on the circumference of the circumcircle. So, the distance between the circumcenter and any of the 3 vertices will be the same and will be equal to the circumradius.