Coordinate Geometry
This topic involves using coordinate geometry formulas to solve problems about the properties of shapes, lengths, and positions on a 2D plane. It's a fundamental skill in ESAT, linking algebraic manipulation with geometric reasoning.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The distance between two points is calculated using Pythagoras' theorem on the right-angled triangle formed by the horizontal and vertical separations.
- The midpoint of a line segment is found by averaging the respective x and y coordinates of its endpoints.
- Always draw a quick sketch of the points and shapes involved. This visual aid helps prevent mistakes and can reveal geometric properties needed to solve the problem.
- Problems often test your knowledge of shapes. Remember properties like parallel sides (equal gradients), perpendicular sides (gradients multiply to -1), and how diagonals behave (e.g., they bisect each other in a parallelogram).
- Coordinate geometry provides a powerful method to prove geometric results algebraically.
Diagram
Formulae
d = √((x2 - x1)2 + (y2 - y1)2) To find the straight-line distance between two points (x1, y1) and (x2, y2), or the length of a line segment.
M = ((x1 + x2)/2, (y1 + y2)/2) To find the coordinates of the midpoint of the line segment connecting points (x1, y1) and (x2, y2).
Definitions
- Midpoint
- The point on a line segment that is equidistant from the two endpoints. Its coordinates are the average of the endpoints' coordinates.
Worked example
The points P(1, 5), Q(9, 1), and R(7, -5) are three vertices of a parallelogram PQRS. Find the coordinates of the fourth vertex S.
- 1
First, sketch the points to visualize the shape.
Since the vertices are given in order PQRS, the diagonal PR must share a midpoint with the diagonal QS.
- 2
Find the midpoint of the diagonal PR.
Let's call it M.
- 3 M = ((1+7)/2, (5+(-5))/2) = (8/2, 0/2) = (4, 0)
- 4
This point M(4, 0) must also be the midpoint of the other diagonal, QS.
Let the coordinates of S be (x, y).
- 5
Using the midpoint formula for QS:
M = ((9+x)/2, (1+y)/2) - 6
Equate the coordinates of M:
4 = (9+x)/2 and 0 = (1+y)/2 - 7
Solve for x:
8 = 9+x ⇒ x = -1 - 8
Solve for y:
0 = 1+y ⇒ y = -1 - 9
Therefore, the coordinates of vertex S are (-1, -1).
Answer: S = (-1, -1)
Common mistakes
- ×Sign errors when subtracting coordinates are the most frequent mistake. Remember that subtracting a negative is equivalent to adding a positive, e.g., 4 - (-2) = 6.
- ×Arithmetic errors during squaring or summing under the square root. Always double-check your calculations, as a small slip can lead to an incorrect final answer.
- ×Ignoring constraints or misinterpreting the question, such as the order of vertices in a polygon (e.g. ABCD vs ABDC), which defines which points form the diagonals.
- ×Confusing the distance and midpoint formulas. The midpoint involves adding and dividing by two (an average), while distance involves subtracting, squaring, and taking a square root (Pythagoras).
No-calculator tips
- ✓When using the distance formula, look for coordinate differences that form a Pythagorean triple (like 3-4-5, 5-12-13, or their multiples like 6-8-10). This lets you find the hypotenuse (the distance) instantly without squaring and summing.
- ✓To simplify a square root, like √(72), find the largest perfect square that divides it. 72 = 36 × 2, so √(72) = √(36) × √(2) = 6*√(2).
- ✓A quick, rough sketch on your paper is invaluable. It can help you spot if your calculated coordinates make sense visually (e.g., if a point should be in the third quadrant but your answer is positive).