Orienting Yourself: The Use of Coordinates · Lesson 5 of 5
Chapter Summary and Practice
“Plot what you know and see whether your answers land in the right place.”
• Recall the language and structure of the Cartesian plane. • Interpret coordinates and quadrant signs confidently. • Apply horizontal, vertical and general distance formulas. • Use coordinates to reason about shapes, reflections and layouts. • Solve mixed coordinate problems with clear justification.
Coordinates turn position into numbers. A point on a plane can be located precisely once we choose two perpendicular reference axes, a common origin and a consistent sign convention. From there, the same numerical description lets us measure distances and reason about shapes.
Why Coordinates Work
A one-dimensional line needs one coordinate. A two-dimensional plane needs two. The ordered pair (x, y) records horizontal position first and vertical position second.
Axes and Origin
The horizontal reference line is the x-axis, the vertical reference line is the y-axis, and their intersection is the origin O(0,0). Right and up are positive directions; left and down are negative.
Points on the Axes
Quadrants
| Quadrant | Coordinate signs |
|---|---|
| I | (+, +) |
| II | (−, +) |
| III | (−, −) |
| IV | (+, −) |
Order Matters
The first coordinate is x and the second is y. In general, (x,y) and (y,x) represent different points. They represent the same point only when x = y.
Distance Along the Axes
Distance Between Any Two Points
The distance formula comes from a right triangle whose legs are the horizontal and vertical coordinate changes. It is therefore an application of the Baudhāyana–Pythagoras theorem.
Coordinate Geometry in Real Layouts
A floor plan can use coordinates to describe walls, doors, furniture and clearances. Points sharing a y-coordinate form horizontal segments, while points sharing an x-coordinate form vertical segments. The method is precise, but a two-dimensional plan does not contain height information.
Further Connections
The chapter also invites us to discover relationships involving midpoints and division of a line segment. If M is the midpoint of S(x₁,y₁) and T(x₂,y₂), then M lies halfway between their corresponding x- and y-coordinates.
The same displacement idea can be extended to trisection: divide the horizontal and vertical changes in the same ratio. This is another example of how coordinates turn geometric movement into arithmetic.
How to Choose the Right Idea
| What you notice | Useful idea |
|---|---|
| Point lies on an axis | One coordinate must be 0 |
| Signs of x and y are given | Identify the quadrant |
| Two points share y | Horizontal distance = |x₂−x₁| |
| Two points share x | Vertical distance = |y₂−y₁| |
| Both coordinates change | Use the distance formula |
| A point is halfway between two endpoints | Average corresponding coordinates |
| A reflected figure is given | Coordinate signs change in a predictable way, while lengths are preserved |
• Writing y before x in an ordered pair, • Forgetting that a point on the x-axis has y = 0, • Using the wrong sign pattern for a quadrant, • Forgetting absolute value for purely horizontal or vertical distance, • Forgetting to square both coordinate differences in the general distance formula, • Assuming a 2-D floor plan contains height information,
Guided Practice
Problem
Where does P(−6,4) lie? What are its perpendicular distances from the axes?
- 1.x is negative and y is positive, so P lies in Quadrant II.
- 2.Its distance from the y-axis is |−6| = 6 units.
- 3.Its distance from the x-axis is |4| = 4 units.
Problem
Three corners of an axis-aligned rectangle are A(2,7), B(8,7) and C(8,3). Find the fourth corner and side lengths.
- 1.AB is horizontal because A and B have the same y-coordinate.
- 2.BC is vertical because B and C have the same x-coordinate.
- 3.The fourth corner must use x = 2 from A and y = 3 from C.
- 4.So D = (2,3).
- 5.AB = 8 − 2 = 6 units and BC = 7 − 3 = 4 units.
Problem
Find the distance between A(−2,3) and B(4,−5).
- 1.Horizontal change = 4 − (−2) = 6.
- 2.Vertical change = −5 − 3 = −8.
- 3.Distance = √(6² + (−8)²).
- 4.Distance = √100 = 10 units.
Problem
Find the midpoint of S(−3,1) and T(5,7).
- 1.Average the x-coordinates: (−3 + 5)/2 = 1.
- 2.Average the y-coordinates: (1 + 7)/2 = 4.
- 3.Therefore the midpoint is M(1,4).
Problem
Points A(1,−8), B(−4,7) and C(−7,−4) are given. Show that they are equally distant from O(0,0).
- 1.OA = √(1² + (−8)²) = √65.
- 2.OB = √((−4)² + 7²) = √65.
- 3.OC = √((−7)² + (−4)²) = √65.
- 4.All three points are the same distance from O.
- 5.Therefore they lie on a circle centred at O with radius √65.
Practice Problems
- What are the coordinates of the point where the two coordinate axes intersect?
- A point W has x-coordinate −4. A point H lies on the vertical line through W. What is the x-coordinate of H, and in which quadrants could H lie?
- Consider R(4,0), A(0,−3), M(−6,−3) and P(−6,3). Without plotting first, identify a horizontal side, a vertical side and a pair of perpendicular sides.
- Plot Z(5,−6). Choose two additional points to form a right triangle and calculate its three side lengths.
- Explain why negative numbers are necessary if every point in a plane is to be represented.
- Check using distances whether M(−3,−4), A(0,0) and G(6,8) are collinear.
- Check using distances whether R(−5,−1), B(−2,−5) and C(4,−13) are collinear.
- Find the midpoint of S(−4,2) and T(8,−6).
- M(−7,1) is the midpoint of A(3,−4) and B(x,y). Find B.
- For A(4,7) and B(16,−2), find the two points that divide AB into three equal parts.
- A circle has centre O(0,0) and radius √65. Decide whether D(−5,6) and E(0,9) lie inside, on or outside the circle.
- A rectangular computer screen uses the bottom-left corner as O(0,0). The screen is 900 units wide and 650 units high. A circular icon has centre (110,160) and radius 70. Does any part lie outside the screen?
- Plot A(2,1), B(−1,2), C(−2,−1) and D(1,−2). Use distances to decide whether ABCD is a square and then find its area.
Quiz
A point with coordinates (−3,5) lies in:
Which form represents every point on the y-axis?
What is the distance between (−2,4) and (5,4)?
When are (x,y) and (y,x) the same point?
The distance formula is based on:
Check that you can explain why two coordinates are needed, read points on both axes and in all four quadrants, keep the order (x,y) correct, identify horizontal and vertical alignments, derive the distance formula from a right triangle, and use coordinates to reason about layouts and geometric shapes.
Key Takeaways
• Coordinates provide a precise numerical language for location. • The Cartesian plane uses two perpendicular axes and the origin as reference. • Ordered pairs record horizontal position first and vertical position second. • Coordinate differences give horizontal and vertical displacement. • The distance formula follows from the Baudhāyana–Pythagoras theorem. • Coordinate methods can solve both practical layout problems and geometric problems.
This completes the chapter. Continue by practising problems where you must decide whether the key idea is location, alignment, midpoint, reflection or distance.
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Distance Between Two Points in the 2-D Plane
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