Coordinate Geometry

What is coordinate geometry?

Coordinate geometry connects algebra with geometry. Instead of describing a point only by its position in a diagram, we assign it two numbers. We can then use calculations to answer geometric questions:

• Where is a point?
• How far apart are two points?
• What point lies halfway between them?
• Do three points form an isosceles or right triangle?
• What coordinates are missing from a geometric construction?

The main idea is:

Represent positions using coordinates, calculate using algebra, and interpret the result geometrically.

The Cartesian coordinate plane

The coordinate plane contains two perpendicular number lines.

The x-axis

The x-axis is horizontal.

• Positive x-values lie to the right.
• Negative x-values lie to the left.

The y-axis

The y-axis is vertical.

• Positive y-values lie above the x-axis.
• Negative y-values lie below the x-axis.

The origin

The axes intersect at the origin: O = (0, 0).

Ordered pairs

A point is written as:

The first coordinate gives horizontal position. The second gives vertical position. For example: P = (3, −2) means:

1. Start at the origin.
2. Move 3 units right.
3. Move 2 units down.

Order matters

The points (3, −2) and (−2, 3) are different. In general:

The two ordered pairs are equal only when the coordinates match in the same positions.

Quadrants and points on the axes

The axes divide the plane into four quadrants.

Important: A point on an axis does not belong to any quadrant. The origin belongs to both axes.

Worked example 1: Classifying points

Classify:

• A = (−4, 3).
• B = (2, −5).
• C = (0, 6).
• D = (−7, 0).

Answers:

• A is in Quadrant II.
• B is in Quadrant IV.
• C lies on the y-axis.
• D lies on the x-axis.

Distance from a point to an axis

For:

the perpendicular distance to the x-axis is:

The perpendicular distance to the y-axis is:

Absolute values are necessary because distance cannot be negative.

Worked example 2

Find the distances from P = (−5, 3) to the coordinate axes. Distance to the x-axis:

Distance to the y-axis:

Distance to the x-axis: 3 units. Distance to the y-axis: 5 units. Remember: Distance to the x-axis uses the y-coordinate, and distance to the y-axis uses the x-coordinate.

Coordinate changes and straight-line distance

Let:

The signed horizontal change from A to B is:

The signed vertical change is:

These changes describe direction as well as size. The corresponding horizontal and vertical lengths are:

Horizontal segment

If the points have the same y-coordinate:

Vertical segment

If the points have the same x-coordinate:

Worked example 3

Find the distance between: A = (−3, 2), B = (5, 2). The segment is horizontal:

Distance: 8 units. For: C = (4, −6), D = (4, 1), the segment is vertical:

Distance: 7 units.

Deriving the distance formula

For two general points, the horizontal and vertical changes form the legs of a right triangle. The straight segment between the points is the hypotenuse. By the Pythagorean theorem:

Therefore:

This is the distance formula.

Important properties

• Distance is nonnegative.
• Reversing the endpoints gives the same distance.
• Distance is zero exactly when the two points are identical.
• The formula gives the straight-line distance, not the distance traveled along a chosen route.

Distance from the origin

For P = (x, y):

Worked example 4: Distance with negative coordinates

Find the distance between: A = (−2, 3), B = (4, −5).

Step 1: Find the coordinate changes

Step 2: Apply the formula

Distance: 10 units.

Verification

The horizontal and vertical lengths are 6 and 8. These form a 6–8–10 right triangle. Sign warning:

Use parentheses when squaring a negative difference.

Exact distances and squared distances

A distance does not always simplify to an integer.

Worked example 5

Find the distance between: A = (1, −2), B = (5, 1).

Now consider: C = (1, 1), D = (5, 3).

Exact distance: 2√5 units. Keep the exact form unless a decimal approximation is requested.

Comparing distances without square roots

Because distances are nonnegative, comparing their squares gives the same ordering. For example:

• If AB² = 20.
• If AC² = 25.

Then AB < AC. This is useful when:

• Comparing side lengths.
• Checking whether a triangle is isosceles.
• Applying the Pythagorean theorem.

Finding a missing coordinate from a distance

Worked example 6

Point P = (x, 2) is 5 units from A = (1, −1). Find all possible values of x.

Step 1: Write the distance equation

Step 2: Square both sides

Step 3: Use both square-root signs

Therefore:

Possible points: (5, 2) and (−3, 2).

Verification

Each point has a horizontal separation of 4 and a vertical separation of 3 from A. Both distances are 5. Key lesson: A fixed-distance condition can produce more than one point.

The midpoint formula

The midpoint lies halfway along a segment. For:

the midpoint is:

Average the x-coordinates and y-coordinates separately.

Why the formula works

Starting from A, move halfway through each coordinate change:

Similarly:

Worked example 7

Find the midpoint of: A = (−4, 7), B = (6, −1).

Midpoint: (1, 3).

Verification

From A to M, the coordinate changes are (5, −4). From M to B, the changes are also (5, −4). The two half-segments have the same length and direction.

Finding a missing endpoint

If M is the midpoint of AB:

Rearrange:

Worked example 8

A = (−3, 5), and the midpoint of AB is M = (2, −1). Find B.

B = (7, −7).

Verification

The midpoint matches the given information.

A midpoint warning

Being equally distant from A and B does not, by itself, make a point their midpoint. A midpoint must also lie on segment AB. For example, (0, 2) is equally distant from (−1, 0) and (1, 0), but their midpoint is (0, 0).

Basic coordinate reflections

Reflections provide a useful way to interpret signs and symmetry.

Worked example 9

Reflect P = (−3, 4):

• Across the x-axis: (−3, −4).
• Across the y-axis: (3, 4).
• Through the origin: (3, −4).
• Across y = x: (4, −3).

Reflections preserve distances.

Reflection through a given midpoint

If A and B are reflections of one another through M, then M is their midpoint. Thus the missing-endpoint formula also finds the reflection of a point through a specified center.

Applications to triangles and rectangles

Worked example 10: Classifying a triangle

Let:

• A = (0, 0).
• B = (4, 0).
• C = (0, 3).

Find the side lengths and classify the triangle.

Since:

the triangle is right-angled. The right angle is at A, where the sides of lengths 3 and 4 meet. Its area is:

Area: 6 square units.

Worked example 11: An isosceles triangle

Let:

• A = (−2, 0).
• B = (2, 0).
• C = (0, 3).

Calculate squared lengths:

Therefore AC = BC. The triangle is isosceles, with equal sides AC and BC. The midpoint of AB is (0, 0), directly below C. The base length is 4 and the perpendicular height is 3. Area: 6 square units.

Worked example 12: An axis-aligned rectangle

Three consecutive vertices of a rectangle are:

• A = (−1, 2).
• B = (5, 2).
• C = (5, −2).

Find the fourth vertex, center, and diagonal length. Because the sides are parallel to the coordinate axes: Fourth vertex: D = (−1, −2). The center is the midpoint of diagonal AC:

Center: (2, 0). The diagonal length is:

Diagonal length: 2√13 units.

Common mistakes and verification

Mistake 1: Reversing the coordinate order

In (x, y), horizontal position comes first.

Mistake 2: Assigning an axis point to a quadrant

If either coordinate is zero, the point is on an axis, not in a quadrant.

Mistake 3: Giving a negative distance

Coordinate changes may be negative. Distances cannot be negative.

Mistake 4: Adding coordinate differences

The expression:

measures the length of a horizontal-and-vertical route, not generally the straight-line distance.

Mistake 5: Forgetting parentheses

For a difference of −7:

Mistake 6: Using differences in the midpoint formula

Midpoints use coordinate sums divided by 2.

Mistake 7: Taking only one square-root sign

From:

both x − 1 = 4 and x − 1 = −4 must be considered.

Mistake 8: Relying on the appearance of a diagram

Unequal axis scales can distort lengths and angles visually. Use the coordinate values and formulas. For a geometrically faithful sketch, use equal unit lengths on both axes.

Verification checklist

• Have I used the same coordinate order throughout?
• Have I handled negative signs correctly?
• Is the distance nonnegative?
• Is the straight-line distance at least the absolute change in either coordinate?
• Does the midpoint lie between the endpoint coordinates?
• Does a recovered endpoint reproduce the given midpoint?
• Have I retained all valid solutions?
• Are lengths in units and areas in square units?