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Geometry Triangles – Circumcentre, Orthocentre

K S Baskar

GRE Geometry Practice Question

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.

Question

A straight line 4x + 3y = 24 forms a triangle with the coordinate axes. What is the distance between the orthocentre and circumcentre of the triangle so formed?
  1. 12 units
  2. 10 units
  3. 5 units
  4. 7 units
  5. 14 units

Explanatory Answer

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

How to determine the points where the line cuts the x and y axes?

The points where the line cuts the x-axis is the x-intercept of the line. The y-coordinate of the x-intercept is zero. So, to find the x-intercept, substitute y = 0 in the equation of the line.
4x + 3(0) = 24 or x = 6.

So, the line cuts the x-axis at (6, 0)

Similarly, to find the coordinates of the point where the line cuts the y-axis (the y-intercept of the line), substitute x = 0 in the equation of the line.
4(0) + 3y = 24 or y = 8.

The line therefore, cuts the y-axis at (0, 8)

It is evident from the diagram that the line and the coordinate axes form a right triangle with sides 6, 8 and 10.

Where is the orthocenter of a right triangle located?

The orthocenter of any triangle is the point of intersection of the altitudes of the triangle.
In a right triangle, the orthocenter lies at the vertex where the right angle is formed.

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

Where is the circumcenter of a right triangle located?

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.

The circumcenter of a right triangle is at located at the mid-point of the hypotenuse.

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)

The coordinates of the orthocenter of this triangle are (0, 0) and those of the circumcenter of this triangle are (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.

Alternative Explanation

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

Why so?

The orthocenter is one of the vertices of the triangle. The circumcenter is the mid-point of the opposite side. A line from a vertex to the mid point of the opposite side is the median to that side.

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

Why so?

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.

And in a right triangle, the circumradius is half of the hypotenuse.
Hence, 5 units.

Why so?

I will leave that part for you to figure out.

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