The point of intersection of the altitudes of a triangle is called its:a)Incentreb)Excentrec)Orthocentred)Centroide)none of theseCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

The three (possibly extended) altitudes intersect in a single point, called the orthocenter of the triangle, usually denoted by H. The orthocenter lies inside the triangle if and only if the triangle is acute (i.e. does not have an angle greater than or equal to a right angle).

Answer

Incenter

Located at intersection of the

angle bisectors

.

Circumcenter

Located at intersection of the perpendicular bisectors of the sides

Centroid

Located at intersection of the medians

Orthocenter

Located at intersection of the

altitudes

Answer

Orthocentre

The point of intersection of the altitudes of a triangle is called its orthocentre. The orthocentre is an important point in a triangle and has several interesting properties.

Definition:
The altitude of a triangle is a line segment drawn from a vertex of the triangle perpendicular to the opposite side (or its extension). The altitudes of a triangle are concurrent, meaning they intersect at a single point. This point of intersection is called the orthocentre.

Properties of Orthocentre:
1. Intersection of altitudes: The orthocentre is the point of intersection of the altitudes of a triangle. This means that if we draw the altitudes from each vertex of the triangle, they will all meet at the orthocentre.

2. Not necessarily inside the triangle: The orthocentre can be located inside, outside, or on the triangle. It depends on the shape and orientation of the triangle.

3. Relationship with acute, obtuse, and right triangles:
- In an acute triangle, the orthocentre is always inside the triangle.
- In an obtuse triangle, the orthocentre is always outside the triangle.
- In a right triangle, the orthocentre coincides with the vertex opposite the right angle.

4. Reflection properties: If we reflect any vertex of the triangle in the opposite side (or its extension), the reflected point lies on the circumcircle of the triangle. The orthocentre is the reflection of each vertex in the opposite side. Hence, the orthocentre also lies on the circumcircle of the triangle.

5. Relationship with other triangle centers: The orthocentre is one of the triangle centers, along with the incenter, circumcenter, and centroid. Each of these centers has unique properties and can be constructed using different elements of the triangle.

Conclusion:
The point of intersection of the altitudes of a triangle is called its orthocentre. It is an important point with several properties related to the altitudes and other triangle centers. Understanding the orthocentre helps in analyzing and solving various problems related to triangles.

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