[FREE] Draw an obtuse scalene triangle and construct its centroid and orthocenter. Which describes the location of

[FREE] Draw an obtuse scalene triangle and construct its centroid and orthocenter. Which describes the location of - brainly.com (2025)

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In an obtuse scalene triangle, the centroid is located inside the triangle, while the orthocenter is found outside of the triangle due to one of its angles being obtuse. The centroid is derived from the intersection of the medians, and the orthocenter comes from the intersection of the altitudes. Therefore, the correct description of their locations is that the centroid is inside and the orthocenter is outside.

Explanation

Understanding the Centroid and Orthocenter in an Obtuse Scalene Triangle

To draw an obtuse scalene triangle, start by choosing three points, ensuring that one angle is greater than 90 degrees (obtuse) and that all sides have different lengths (scalene). For example, vertices A, B, and C can be positioned such that angle ABC is obtuse.

Next, construct the centroid (G) and orthocenter (H) of the triangle.

Finding the Centroid

The centroid of a triangle is the intersection of its three medians. A median is a line drawn from a vertex to the midpoint of the opposite side. In triangle ABC, find the midpoints of sides BC, AC, and AB. Draw the medians from vertices A, B, and C to these midpoints. The point where all three medians intersect is the centroid G, which is always located inside the triangle.

Finding the Orthocenter

The orthocenter is the intersection of the triangle's altitudes. An altitude is a perpendicular drawn from a vertex to the opposite side. In an obtuse triangle, the orthocenter H lies outside the triangle since one of the sides must extend outside the other two vertices. Therefore, for the obtuse scalene triangle constructed, the orthocenter will be outside the triangle.

Based on this construction, we can conclude that:

The centroid is always inside the triangle.
The orthocenter in an obtuse triangle is always outside the triangle.

Thus, the correct statement regarding the location of the points of concurrency is:

The centroid is inside the triangle and the orthocenter is outside of the triangle.

Learn more about Triangle Concurrency here:

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In an obtuse scalene triangle, the centroid is always located inside the triangle, while the orthocenter is found outside due to the obtuse angle. Thus, the correct answer is option B: the centroid is inside the triangle, and the orthocenter is outside of the triangle.

Explanation

To draw an obtuse scalene triangle, start by selecting three points in such a way that one angle is greater than 90 degrees and all sides are of different lengths. For example, label the vertices A, B, and C, with angle ABC being obtuse.

Next, construct the centroid (G) and orthocenter (H) of the triangle.

Finding the Centroid

The centroid is located at the intersection of the three medians of the triangle. A median is a line segment drawn from a vertex to the midpoint of the opposite side. Therefore:

Find the midpoints of each side:
- D is the midpoint of BC,
- E is the midpoint of AC,
- F is the midpoint of AB.
Draw line segments (medians) from each vertex (A, B, C) to the midpoints (D, E, F).
The meeting point of these three medians is the centroid G, which is always located inside the triangle.

Finding the Orthocenter

The orthocenter is the intersection of the triangle's altitudes. An altitude is a line drawn from a vertex that is perpendicular to the opposite side. In an obtuse triangle:

Drop a perpendicular from the vertex opposite the obtuse angle to the line extended from the side opposite this vertex.
Draw the altitudes from the other two vertices to their respective opposite sides.
The point where all three altitudes meet is the orthocenter H. For obtuse triangles, the orthocenter is outside the triangle.

Conclusion

From this construction, we find that:

The centroid is located inside the triangle.
The orthocenter is located outside the triangle.

Therefore, the correct option that describes the locations of the points of concurrency is:

B. The centroid is inside the triangle, and the orthocenter is outside of the triangle.

Examples & Evidence

For example, consider a triangle with vertices at (0, 0), (4, 1), and (1, 3). This forms an obtuse triangle. The centroid can be calculated as the average of the coordinates of these vertices, while the orthocenter can be found by drawing perpendicular lines from each vertex to the opposite side and identifying their intersection point.

In triangle geometry, it is known that for any obtuse triangle, the centroid will always be located inside the triangle, while the orthocenter extends out due to the nature of obtuse angles. This can be confirmed through geometric proofs and constructions.

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