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 = (\frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3})

Front

If point P is the centroid of triangle ABC, and PL = 18, what is the length of BL if BP = 10?

Back

BL = 2 * PL = 2 * 18 = 36.

Front

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

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 the length of AC in triangle ABC is 22, what is the length of AL?

Back

AL = \frac{AC}{2} = \frac{22}{2} = 11.

Front

How do you determine the length of PL if BP = 10 and P is the centroid?

Back

Since P is the centroid, PL = \frac{1}{3} * BP = \frac{1}{3} * 10 = 5.

Front

What is the formula to calculate the area of a triangle given its vertices?

Back

The area A can be calculated using the formula: A = \frac{1}{2} |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|.

Front

If the coordinates of the vertices of a triangle are (3,1), (9,3), and (6,1), what are the coordinates of the centroid?

Back

The centroid coordinates are G = (\frac{3 + 9 + 6}{3}, \frac{1 + 3 + 1}{3}) = (6, 1).

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 coordinates of a triangle's vertices are (5, -4), (1, 2), and (3, 0), what is the centroid?

Back

The centroid is G = (\frac{5 + 1 + 3}{3}, \frac{-4 + 2 + 0}{3}) = (3, -2/3).