Understanding the Orthocentre of a Right-Angled Triangle

The orthocentre of a triangle is a fundamental geometric point. It is defined as the intersection point of the altitudes of the triangle. An altitude from a vertex is the perpendicular line segment drawn from that vertex to the opposite side (or its extension).

Locating the Orthocentre in a Right-Angled Triangle

Let's consider a right-angled triangle, say \(\triangle \text{ABC}\), with the right angle at vertex \(\text{B}\). To find the orthocentre, we need to determine the location where its three altitudes intersect.

• Altitude from vertex A: The altitude from vertex \(\text{A}\) must be perpendicular to the opposite side \(\text{BC}\). Since \(\triangle \text{ABC}\) is right-angled at \(\text{B}\), the side \(\text{AB}\) is already perpendicular to \(\text{BC}\). Therefore, the altitude from \(\text{A}\) is the side \(\text{AB}\) itself.
• Altitude from vertex C: Similarly, the altitude from vertex \(\text{C}\) must be perpendicular to the opposite side \(\text{AB}\). Since the angle at \(\text{B}\) is 90 degrees, the side \(\text{BC}\) is already perpendicular to \(\text{AB}\). Therefore, the altitude from \(\text{C}\) is the side \(\text{BC}\) itself.
• Altitude from vertex B: The altitude from vertex \(\text{B}\) must be perpendicular to the opposite side \(\text{AC}\) (the hypotenuse). This altitude is a line segment drawn from \(\text{B}\) that meets \(\text{AC}\) at a 90-degree angle.

The orthocentre is the point where these three altitudes intersect. We can see that the altitude from \(\text{A}\) (which is \(\text{AB}\)) and the altitude from \(\text{C}\) (which is \(\text{BC}\)) both intersect at vertex \(\text{B}\). Since all three altitudes must intersect at the same point (the orthocentre), the altitude from \(\text{B}\) to \(\text{AC}\) must also pass through \(\text{B}\).

Thus, the orthocentre of a right-angled triangle is located exactly at the vertex where the right angle is formed. This vertex is a point that lies on the triangle itself (specifically, it's one of the vertices).

Analyzing the Options for Orthocentre Location

Based on our understanding, let's examine the given options regarding the orthocentre of a right-angled triangle:

• Option 1: Its orthocentre lies inside the triangle. This is true for acute-angled triangles, where all angles are less than 90 degrees. It is not true for a right-angled triangle.
• Option 2: Its orthocentre lies outside the triangle. This is true for obtuse-angled triangles, where one angle is greater than 90 degrees. It is not true for a right-angled triangle.
• Option 3: Its orthocentre lies on the triangle. As we've determined, the orthocentre of a right-angled triangle is located at the vertex with the right angle. A vertex is considered a point on the triangle. Therefore, this statement is correct.
• Option 4: It has no orthocentre. Every triangle, regardless of its type (acute, right, obtuse), has a unique orthocentre where its altitudes intersect. This statement is incorrect.

Therefore, the correct statement is that the orthocentre of a right-angled triangle lies on the triangle, specifically at the vertex containing the right angle.

Revision Table: Orthocentre Location by Triangle Type

Additional Information: Other Triangle Centres

Besides the orthocentre, triangles have other significant centres:

• Centroid: The intersection point of the medians. A median connects a vertex to the midpoint of the opposite side. The centroid is always inside the triangle.
• Circumcentre: The intersection point of the perpendicular bisectors of the sides. It is the centre of the circumcircle, which passes through all three vertices. Its location varies (inside for acute, on the hypotenuse midpoint for right, outside for obtuse).
• Incentre: The intersection point of the angle bisectors. It is the centre of the incircle, which is tangent to all three sides. The incentre is always inside the triangle.

Each centre has unique properties and is found by intersecting different types of lines associated with the triangle.