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The orthocentere of an obtuse angled triangle lies
- a)on the greatest side of the triangle
- b)outside the triangle
- c)on the smallest side of the triangle
- d)inside the triangle
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
The orthocentere of an obtuse angled triangle liesa)on the greatest si...
Understanding the Orthocenter
The orthocenter of a triangle is the point where the three altitudes intersect. Its position varies depending on the type of triangle—acute, right, or obtuse.
Orthocenter in Obtuse-Angled Triangles
In an obtuse-angled triangle, one angle is greater than 90 degrees. This unique property influences the location of the orthocenter.
Key Points on the Position of the Orthocenter
- **Position Outside the Triangle**:
- In an obtuse triangle, the orthocenter is located outside the triangle. This occurs because the altitude from the vertex with the obtuse angle extends beyond the triangle, intersecting the extensions of the other two sides.
- **Geometric Implication**:
- Since one angle is greater than 90 degrees, the perpendicular dropped from that vertex to the opposite side does not intersect within the triangle, pushing the orthocenter outside.
Comparative Analysis with Other Triangles
- **Acute Triangle**:
- The orthocenter is located inside, as all angles are less than 90 degrees, allowing all altitudes to intersect within the triangle.
- **Right Triangle**:
- The orthocenter coincides with the vertex of the right angle, which is also inside the triangle.
Conclusion
The correct answer is option 'B': the orthocenter of an obtuse-angled triangle lies outside the triangle, which can be visually confirmed through geometric constructions and properties of triangle altitudes. Understanding these properties is essential for solving geometric problems in bank exams and other competitive assessments.
The orthocenter of a triangle is the point where the three altitudes intersect. Its position varies depending on the type of triangle—acute, right, or obtuse.
Orthocenter in Obtuse-Angled Triangles
In an obtuse-angled triangle, one angle is greater than 90 degrees. This unique property influences the location of the orthocenter.
Key Points on the Position of the Orthocenter
- **Position Outside the Triangle**:
- In an obtuse triangle, the orthocenter is located outside the triangle. This occurs because the altitude from the vertex with the obtuse angle extends beyond the triangle, intersecting the extensions of the other two sides.
- **Geometric Implication**:
- Since one angle is greater than 90 degrees, the perpendicular dropped from that vertex to the opposite side does not intersect within the triangle, pushing the orthocenter outside.
Comparative Analysis with Other Triangles
- **Acute Triangle**:
- The orthocenter is located inside, as all angles are less than 90 degrees, allowing all altitudes to intersect within the triangle.
- **Right Triangle**:
- The orthocenter coincides with the vertex of the right angle, which is also inside the triangle.
Conclusion
The correct answer is option 'B': the orthocenter of an obtuse-angled triangle lies outside the triangle, which can be visually confirmed through geometric constructions and properties of triangle altitudes. Understanding these properties is essential for solving geometric problems in bank exams and other competitive assessments.
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