# Midpoint

#### Examples

Calculate the coordinates of the midpoint from the line segment AB.

Calculate the coordinates of Point C in the line segment AC, knowing that the midpoint is B = (2, −2) and an endpoint is A = (−3, 1).

If M_{1} = (2, 1), M_{2 } = (3, 3) and M_{3} = (6, 2) are the midpoints of the sides that make up a triangle, what are the coordinates of the vertices?

x_{1} = 7 x_{5} = 7 x_{3} = −1

y_{1} = 4 y_{5} = 0 y_{3} = 3

A(7, 4)B(5, 0) C(−1, 2)

If the line segment AB with endpoints A = (1,3) and B = (7, 5) is divided into four equal parts, what are the coordinates of the points of division?

### Symmetric Point

If **A'** is the **symmetric** of **A** with respect to **M**, then **M** is the **midpoint** of the line segment** AA'**.

Calculate the symmetric point of A = (7, 4) with the midpoint M = (3, −11).

#### Examples

Calculate the symmetric point of A = (4, −2) for midpoint M = (2, 6).

### Three Collinear Points

#### Examples

Determine whether A = (−2, −3), B = (1, 0) and C = (6, 5) are alligned points.

Calculate the value of **a** in the following aligned points.

### Centroid Coordinates

#### Examples

Given the vertices of a triangle A = (−3, −2), B = (7, 1) and C = (2, 7), calculate the coordinates of the centroid.

If two vertices of a triangle are A = (2, 1) and B = (1, 0) and the centroid is G = (2/3, 0), calculate the third vertex.

### Dividing a Segment

Dividing the segment AB at a given ratio r is to determine a point P on the line containing the segment AB, so that both sides PA and PB have a ratio of:

#### Examples

Calculate the points P and Q that divide the line segment with endpoints A = (-1, -3) and B = (5, 6) into three equal parts?

The coordinates of the endpoints from the line segment AB are: A = (2, −1) and B = (8, −4). Find the coordinates of point C such that it divides the line segment into two equal parts.

Given the points A (3, 2) and B (5, 4), find a point C such that it is aligned with A and B and a ratio of is obtained.