6.3 Centroid

Front

What is the centroid of 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 and is located at the average of the vertices' coordinates.

Front

How do you calculate 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 calculated using the formula: G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right).

Front

If the vertices of a triangle are (2, 3), (4, 5), and (6, 7), what are the coordinates of the centroid?

Back

The centroid is G = \left(\frac{2 + 4 + 6}{3}, \frac{3 + 5 + 7}{3}\right) = (4, 5).

Front

What is the relationship between the centroid and the lengths of the segments created by the centroid on the medians?

Back

The centroid divides each median into two segments, with the segment connecting the vertex to the centroid being twice as long as the segment connecting the centroid to the midpoint of the opposite side.

Front

If point P is the centroid of triangle ABC and PL = 18, what is the length of BP?

Back

BP = 36, since the centroid divides the median in a 2:1 ratio.

Front

If BP = 10, what is the length of BL if P is the centroid?

Back

BL = 15, as the centroid divides the median in a 2:1 ratio.

Front

What is the formula for the length of a median in a triangle with vertices (x1, y1), (x2, y2), (x3, y3)?

Back

The length of the median from vertex A to side BC can be calculated using the formula: m_a = \sqrt{\frac{2b^2 + 2c^2 - a^2}{4}}, where a, b, and c are the lengths of the sides.

Front

What is the significance of the centroid in real-world applications?

Back

The centroid is used in various fields such as engineering, architecture, and physics to determine the center of mass and balance of structures.

Front

How does the centroid relate to the area of a triangle?

Back

The centroid is located at a point that divides the triangle into three smaller triangles of equal area.

Front

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

Back

The centroid is G = \left(\frac{1 + 3 + 5}{3}, \frac{2 + 4 + 6}{3}\right) = (3, 4).