6.3 Centroid

Front

What is the definition of a centroid in a triangle?

Back

The centroid of a triangle is the point where the three medians intersect. It is also the center of mass of the triangle.

Front

How do you find the coordinates of the centroid of a triangle with vertices (x1, y1), (x2, y2), (x3, y3)?

Back

The coordinates of the centroid (G) can be found using the formula: G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right).

Front

If point P is the centroid of triangle ABC, and the length of segment AP is 6, what is the length of segment PB?

Back

The length of segment PB is also 6, since the centroid divides each median into a ratio of 2:1.

Front

What is the ratio in which the centroid divides each median of a triangle?

Back

The centroid divides each median in the ratio 2:1, with the longer segment being closer to the vertex.

Front

If the coordinates of the vertices of a triangle are (1, 2), (3, 4), and (5, 6), what are the coordinates of the centroid?

Back

The coordinates of the centroid are \left(\frac{1+3+5}{3}, \frac{2+4+6}{3}\right) = (3, 4).

Front

In triangle ABC, if the lengths of the medians are 12, 16, and 20, what is the length of the median from vertex A?

Back

The length of the median from vertex A is 12.

Front

What is the significance of the centroid in a triangle?

Back

The centroid is the balance point of the triangle, where it would balance perfectly if made of a uniform material.

Front

If the centroid of triangle ABC is at (4, 5), what does this indicate about the distribution of the triangle's area?

Back

It indicates that the area of the triangle is evenly distributed around the point (4, 5).

Front

How does the position of the centroid change if the vertices of the triangle are moved?

Back

The position of the centroid will change according to the new coordinates of the vertices, as it is calculated based on their average.

Front

If the lengths of segments from the centroid to the vertices of a triangle are 3, 4, and 5, what is the total length of the medians?

Back

The total length of the medians is 3 + 4 + 5 = 12.