Class 10 Exam > Class 10 Questions > In an equilateral triangle, the incentre, cir...
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In an equilateral triangle, the incentre, circumcentre, orthocentre and centroid are:
- a)concylic
- b)coincident
- c)collinear
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
In an equilateral triangle, the incentre, circumcentre, orthocentre an...
The centroid is the intersection of the three medians while the incentre is the intersection of the three (internal) angle bisectors. In an equilateral triangle, each median is also an angle bisector (and vice versa), the centroid coincides with the incentre. In fact, the centroid, incentre, circumcentre and orthocentre of an equilateral triangle are coincide at the same point.
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In an equilateral triangle, the incentre, circumcentre, orthocentre an...
Correct answet is b
Justification:
The centroid is the intersection of the three medians while the incentre is the intersection of the three (internal) angle bisectors. In an equilateral triangle, each median is also an angle bisector (and vice versa), so the centroid coincides with the incentre. In fact, the centroid, incentre, circumcentre and orthocentre of an equilateral triangle are coincide at the same point .
Justification:
The centroid is the intersection of the three medians while the incentre is the intersection of the three (internal) angle bisectors. In an equilateral triangle, each median is also an angle bisector (and vice versa), so the centroid coincides with the incentre. In fact, the centroid, incentre, circumcentre and orthocentre of an equilateral triangle are coincide at the same point .
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In an equilateral triangle, the incentre, circumcentre, orthocentre an...
Equilateral Triangle Centers
In an equilateral triangle, several important points coincide at the same location. These points include the incentre, circumcentre, orthocentre, and centroid.
Key Points of Coincidence
- **Incentre**: The point where the angle bisectors of the triangle intersect and is the center of the inscribed circle.
- **Circumcentre**: The point where the perpendicular bisectors of the sides intersect, serving as the center of the circumscribed circle.
- **Orthocentre**: The intersection point of the altitudes of the triangle. In an equilateral triangle, it coincides with the centroid.
- **Centroid**: The point where the medians of the triangle intersect, and it represents the triangle's center of mass.
Explanation of Coincidence
In an equilateral triangle, all sides and angles are equal. This symmetry causes the following:
- Since the triangle is equilateral, the angle bisectors, perpendicular bisectors, altitudes, and medians all coincide at the same point.
- Therefore, the incentre, circumcentre, orthocentre, and centroid are not just collinear or concyclic; they are actually the same point.
Conclusion
Hence, the correct answer to the question is option 'B': the incentre, circumcentre, orthocentre, and centroid of an equilateral triangle are coincident, meaning they occupy the same position in space.
In an equilateral triangle, several important points coincide at the same location. These points include the incentre, circumcentre, orthocentre, and centroid.
Key Points of Coincidence
- **Incentre**: The point where the angle bisectors of the triangle intersect and is the center of the inscribed circle.
- **Circumcentre**: The point where the perpendicular bisectors of the sides intersect, serving as the center of the circumscribed circle.
- **Orthocentre**: The intersection point of the altitudes of the triangle. In an equilateral triangle, it coincides with the centroid.
- **Centroid**: The point where the medians of the triangle intersect, and it represents the triangle's center of mass.
Explanation of Coincidence
In an equilateral triangle, all sides and angles are equal. This symmetry causes the following:
- Since the triangle is equilateral, the angle bisectors, perpendicular bisectors, altitudes, and medians all coincide at the same point.
- Therefore, the incentre, circumcentre, orthocentre, and centroid are not just collinear or concyclic; they are actually the same point.
Conclusion
Hence, the correct answer to the question is option 'B': the incentre, circumcentre, orthocentre, and centroid of an equilateral triangle are coincident, meaning they occupy the same position in space.
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