Finding Coordinates from Geometrical Information
Coordinate geometry links algebra with shape properties. Knowing how to use given geometrical information allows you to find unknown coordinates accurately on a grid.
Midpoint of a Line
The midpoint of a line segment is the point halfway between two endpoints.
If the endpoints are:
\( (x_1, y_1) \) and \( (x_2, y_2) \)
the midpoint is:
$$
\left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)
$$
This works by averaging the x coordinates and averaging the y coordinates.
Example
Endpoints are:
\( (2, 5) \) and \( (8, 1) \)
Midpoint is:
$$
\left(\frac{2 + 8}{2},\ \frac{5 + 1}{2}\right)
$$
$$
(5,\ 3)
$$
Fourth Vertex of a Parallelogram
A parallelogram has:
• opposite sides parallel
• opposite sides equal in length
This means the shape can be translated using vectors.
If three vertices are known, the fourth can be found by adding or subtracting coordinates.
One common approach is to use the fact that diagonals of a parallelogram bisect each other.
If \( A, B, C, D \) are vertices in order, then the midpoint of diagonal \( AC \) equals the midpoint of diagonal \( BD \).
So:
Midpoint of \( AC \) equals midpoint of \( BD \)
Example structure
If points are \( A \), \( B \) and \( C \) with \( A \) adjacent to \( B \) and \( C \), the fourth vertex \( D \) is found using:
$$
D = B + C - A
$$
This rule works because the movement from \( A \) to \( B \) is repeated from \( C \) to \( D \).
The order of points must match the shape correctly
Location Determined by Distance and Angle
Sometimes a point is located using:
• a given distance from a known point
• an angle made with a given line
This is often done by constructing a right angled triangle using trigonometry.
If the distance from a point is \( r \) and the angle to a horizontal line is \( \theta \), the horizontal and vertical changes can be found.
Horizontal change:
$$
r\cos\theta
$$
Vertical change:
$$
r\sin\theta
$$
These changes are then added to the known coordinates.
Example structure
Starting point is \( (x, y) \). The new point is:
$$
(x + r\cos\theta,\ y + r\sin\theta)
$$
The signs may change depending on the direction of movement.
Using a given line that is not horizontal means you must consider the angle relative to that line and draw an accurate diagram.
Clear diagrams help identify the correct direction of the coordinate change
Key Points to Remember
The midpoint is found by averaging x coordinates and averaging y coordinates.
Parallelogram vertices can be found using coordinate addition and subtraction.
Diagonals of a parallelogram share the same midpoint.
Distance and angle problems often use sine and cosine to find coordinate changes.
A correct diagram helps ensure signs and directions are correct.
Finding coordinates from geometrical information combines shape properties with algebra and allows locations to be calculated precisely on a coordinate grid.