What is the formula to find the coordinates of a point that divides a line segment in the ratio m₁:m₂?

Coordinates of P = ((m₁ × x₂ + m₂ × x₁) / (m₁ + m₂), (m₁ × y₂ + m₂ × y₁) / (m₁ + m₂))

If point P divides the line segment from A(2, 3) to B(8, 9) in the ratio 1:3, what are the coordinates of point P?

P = ((1 × 8 + 3 × 2) / (1 + 3), (1 × 9 + 3 × 3) / (1 + 3)) = (6.5, 7.5)

True or False: The midpoint of a line segment defined by points A(x₁, y₁) and B(x₂, y₂) is given by ((x₁ + x₂) / 2, (y₁ + y₂) / 2).

True

What is the centroid of a triangle with vertices A(1, 2), B(4, 6), and C(7, 8)?

Centroid = ((1 + 4 + 7) / 3, (2 + 6 + 8) / 3) = (4, 5.33)

Fill in the blank: The centroid divides each median of a triangle in the ratio ___.

2:1

How do you find the coordinates of the trisection points on a line segment from A(0, 0) to B(9, 9)?

Coordinates of trisection points:

• P₁ = (3, 3) point

• Q₂ = (6, 6) point

• Divides segment into thirds

What is the equation of the line passing through points (3, 4) and (7, 8)?

Slope m = (8 - 4) / (7 - 3) = 1; Equation: y - 4 = 1(x - 3) or y = x + 1

True or False: The coordinates of the midpoint of the line segment from A(-1, -1) to B(3, 3) are (1, 1).

False; The midpoint is (1, 1) but the correct calculation is ((-1 + 3) / 2, (-1 + 3) / 2) = (1, 1).

If point P(5, 5) divides the segment from A(0, 0) to B(x, y) in a ratio of 2:3, how do you find x and y?

Set up equations: 5 = (2x + 3 × 0) / (2 + 3) and 5 = (2y + 3 × 0) / (2 + 3); Solve for x and y to find x = 7.5 and y = 7.5.

Calculate the coordinates of the centroid of a triangle with vertices at A(2, 3), B(4, 5), and C(6, 7).

Centroid = ((2 + 4 + 6) / 3, (3 + 5 + 7) / 3) = (4, 5)

Fill in the blank: The section formula can be used to find a point that divides a line segment into a ratio of ___ and ___.

M₁ and m₂

What is the distance between points A(1, 1) and B(4, 5)?

Distance = √((4 - 1)² + (5 - 1)²) = √(3² + 4²) = √(9 + 16) = √25 = 5

True or False: The coordinates of the centroid are calculated by averaging the coordinates of the triangle's vertices.

True

What are the coordinates of the point that trisects the segment from A(-3, 2) to B(3, -2)?

Coordinates

Trisecting segment from A to B

• P₁ at (-1, 1)

• P₂ at (1, -2)

• Divides segment into thirds

What is the formula to find the coordinates of a point P that divides a line segment AB in the ratio m1:m2?

Coordinates formula for point P.

• Uses ratio m1:m2 for division.

• Calculates x and y coordinates.

• A and B are endpoints specified.