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Where does the orthocentre of an obtuse angled triangle lie with respect to the triangle?

Difficulty: Easy

Correct Answer: outside the triangle

Explanation:


Introduction / Context:

This conceptual geometry question tests your knowledge of the orthocentre, which is the point of intersection of the altitudes of a triangle. The location of the orthocentre depends on the type of triangle: acute, right angled, or obtuse angled.



Given Data / Assumptions:

  • We are dealing with an obtuse angled triangle, which has one angle greater than 90 degrees.
  • The orthocentre is defined as the intersection point of the three altitudes of the triangle.
  • We are asked to describe where this orthocentre lies relative to the triangle.


Concept / Approach:

The position of the orthocentre varies with the nature of the triangle. In an acute triangle, all angles are less than 90 degrees, and the orthocentre lies inside. In a right angled triangle, the orthocentre coincides with the right angled vertex. In an obtuse triangle, at least one altitude must be drawn outside the triangle by extending a side, and the intersection of the altitudes therefore lies outside the triangle.



Step-by-Step Solution:

Step 1: Recall that an obtuse angled triangle has exactly one angle greater than 90 degrees.Step 2: Draw a rough sketch of an obtuse triangle and drop the perpendicular (altitude) from the vertex at the obtuse angle to the opposite side.Step 3: Notice that this perpendicular falls on the extension of the opposite side, not on the segment itself, so part of the construction lies outside the triangle.Step 4: Similarly, at least one of the other altitudes will also be drawn outside the triangle in extended form.Step 5: The point where the three altitudes intersect is therefore outside the original triangle.


Verification / Alternative check:

Compare this with the known facts: in an acute triangle, all angles are small, so altitudes intersect inside. In a right angled triangle, two altitudes lie along sides and meet at the right angle vertex. This pattern confirms that only for an obtuse angled triangle does the orthocentre fall outside.



Why Other Options Are Wrong:

Inside the triangle refers to the case for acute triangles, not obtuse triangles. On one side or on a vertex are special positions for right angled triangles. The option none of these is incorrect because one of the specific given statements, outside the triangle, is correct.



Common Pitfalls:

Many learners confuse the orthocentre with the circumcentre or centroid, whose positions behave differently. Others assume all special points must lie inside the triangle, which is not true for obtuse triangles.



Final Answer:

For an obtuse angled triangle, the orthocentre lies outside the triangle.


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