why is the centroid always inside the triangle?
bcz it is. Dont question the system boi.
lol this got asked 5 years ago, no answers.
The centroid of a triangle is always located inside the triangle because it is the point of intersection of the medians. A median of a triangle is a line segment connecting a vertex of the triangle to the midpoint of the opposite side.
To understand why the centroid is always inside the triangle, you can follow these steps:
1. Draw an arbitrary triangle on a piece of paper.
2. Identify one of the vertices of the triangle, and draw a line segment connecting it to the midpoint of the opposite side. This line segment is a median.
3. Repeat step 2 for the other two vertices of the triangle, creating the other two medians.
4. The three medians will intersect at a single point inside the triangle. This point is called the centroid.
Now, let's explain why the centroid is always inside the triangle:
1. By construction, each median passes through the midpoint of the opposite side. The midpoint divides the side into two equal segments.
2. Since the centroid is the point of intersection of the medians, it is also the point where the three medians split each other into equal segments.
3. Consider any two medians that intersect at the centroid. The segment of each median between the centroid and the opposite vertex is shorter than the segment between the centroid and any other point on the opposite side.
4. By the same logic, this is true for the other two pairs of medians and their intersection with the centroid.
5. Therefore, the centroid is the point that "balances" the lengths of all three medians, making it the center of gravity of the triangle.
6. As a result of this balancing act, the centroid is guaranteed to be located inside the triangle.
So, in summary, the centroid of a triangle is always inside the triangle because it is the point where the medians, which connect each vertex to the midpoint of the opposite side, intersect.