Coordinate Geometry. Grade 7 Math: Coordinate Geometry Subtopic Navigator Introduction: The Coordinate Plane Understanding the Coordinate System Plotting Points in All Four Quadrants Reading Coordinates from Graphs Distance Between Points Reflections Across Axes Finding Missing Coordinates Real-World Applications Geometric Shapes on Coordinate Plane Practice Exercises Conclusion Learning Objectives Understand and use the Cartesian coordinate system Plot points accurately in all four quadrants Read coordinates from graphs and diagrams Calculate horizontal and vertical distances between points Reflect points across x-axis and y-axis Find missing coordinates of geometric shapes Apply coordinate geometry to real-world situations Plot and analyze polygons on the coordinate plane Introduction: The Coordinate Plane Coordinate geometry, also called Cartesian geometry, is a system that uses numbers to represent points on a plane. This system was developed by French mathematician René Descartes and allows us to precisely locate points using pairs of numbers called coordinates. Key Terms: Coordinate Plane: A two-dimensional surface formed by two perpendicular number lines x-axis: The horizontal number line (left-right direction) y-axis: The vertical number line (up-down direction) Origin: The point (0,0) where the x-axis and y-axis intersect Ordered Pair: A pair of numbers (x,y) that gives a point's location Quadrant: One of the four regions created by the intersection of the axes Quadrant II (-,+) | Quadrant I (+,+) ------------------|------------------ Quadrant III (-,-) | Quadrant IV (+,-) The coordinate plane is divided into four quadrants by the x-axis and y-axis. Quadrant I is in the upper right, Quadrant II in the upper left, Quadrant III in the lower left, and Quadrant IV in the lower right. The Coordinate System Every point on the coordinate plane is identified by an ordered pair (x,y). The first number (x-coordinate) tells how far to move horizontally from the origin. The second number (y-coordinate) tells how far to move vertically. Reading Coordinates: (x, y) = (horizontal position, vertical position) Positive x: move RIGHT from origin Negative x: move LEFT from origin Positive y: move UP from origin Negative y: move DOWN from origin Example 1: Understanding Positive and Negative Coordinates Explanation: Point A: (3, 2) → Move 3 units RIGHT, 2 units UP (Quadrant I) Point B: (-4, 5) → Move 4 units LEFT, 5 units UP (Quadrant II) Point C: (-2, -3) → Move 2 units LEFT, 3 units DOWN (Quadrant III) Point D: (5, -1) → Move 5 units RIGHT, 1 unit DOWN (Quadrant IV) Point E: (0, 4) → On y-axis (no horizontal movement) Point F: (-3, 0) → On x-axis (no vertical movement) Example 2: Quadrant Identification Identify which quadrant each point is in: A(2, 3), B(-5, 1), C(-4, -2), D(3, -5) Solution: A(2, 3): x=2 (positive), y=3 (positive) → Quadrant I B(-5, 1): x=-5 (negative), y=1 (positive) → Quadrant II C(-4, -2): x=-4 (negative), y=-2 (negative) → Quadrant III D(3, -5): x=3 (positive), y=-5 (negative) → Quadrant IV Common Mistake: Incorrect: Reading (x,y) as (y,x) - switching the coordinates Correct: Always remember: x comes first, then y Mnemonic: "x is a cross (horizontal), y to the sky (vertical)" Practice Questions What are the coordinates of the origin? In which quadrant would you find the point (-3, 4)? Describe how to get to the point (5, -2) from the origin. If a point has coordinates (0, -6), on which axis does it lie? Explain the difference between x-coordinate and y-coordinate. Plotting Points in All Four Quadrants Plotting points accurately requires careful attention to both the sign and magnitude of coordinates. Always start at the origin and move horizontally first, then vertically. Steps for Plotting Points: 1. Start at the origin (0,0) 2. Move horizontally: RIGHT if x is positive, LEFT if x is negative 3. Move vertically: UP if y is positive, DOWN if y is negative 4. Mark the point with a dot and label it 5. Double-check your coordinates Example 1: Plotting Points in Different Quadrants Plot these points: P(2, 3), Q(-4, 1), R(-3, -2), S(5, -4) Solution Steps: Point P(2, 3): - Start at origin (0,0) - Move 2 units RIGHT (positive x) - Move 3 units UP (positive y) - Mark and label point P Point Q(-4, 1): - Start at origin (0,0) - Move 4 units LEFT (negative x) - Move 1 unit UP (positive y) - Mark and label point Q Point R(-3, -2): - Start at origin (0,0) - Move 3 units LEFT (negative x) - Move 2 units DOWN (negative y) - Mark and label point R Point S(5, -4): - Start at origin (0,0) - Move 5 units RIGHT (positive x) - Move 4 units DOWN (negative y) - Mark and label point S Example 2: Text-based Coordinate Grid Here's a simple text representation of a coordinate grid: y-axis ↑ 4 | ·B ·A 3 | 2 | 1 | ·C ·D 0 +-----------→ x-axis -1 | ·E -2 | ·F -3 | -4 | Key: A(3,4), B(-2,4), C(-2,1), D(1,1), E(1,-1), F(-2,-2) Point Coordinates Movement from Origin Quadrant A (4, 2) Right 4, Up 2 I B (-3, 5) Left 3, Up 5 II C (-2, -4) Left 2, Down 4 III D (6, -3) Right 6, Down 3 IV E (0, 5) No horizontal, Up 5 y-axis Principles Practice Plot the point (-5, 3) on a coordinate plane (describe the steps). Plot the point (4, -2) on a coordinate plane (describe the steps). Which point would be further from the origin: (6, 0) or (0, -5)? Plot three points that would form a right triangle with the origin. What is special about points that lie exactly on the x-axis? Reading Coordinates from Graphs Being able to read coordinates from graphs is just as important as plotting points. This skill requires understanding scale and being able to determine exact coordinates from a visual representation. Reading Coordinates from a Graph: 1. Locate the point on the graph 2. Draw an imaginary vertical line to the x-axis 3. Read the x-coordinate where this line hits the x-axis 4. Draw an imaginary horizontal line to the y-axis 5. Read the y-coordinate where this line hits the y-axis 6. Write as (x, y) Example 1: Reading Coordinates from a Diagram Consider this text-based coordinate grid with points A through F: 5 | ·B 4 | ·A 3 | 2 | ·C 1 | 0 +-----------→ -1 | ·D -2 | ·E -3 | ·F -4 | -4 -3 -2 -1 0 1 2 3 4 5 Determine coordinates: Point A: From A, go down to x-axis: x=3, go left to y-axis: y=4 → A(3,4) Point B: From B, go down to x-axis: x=2, go left to y-axis: y=5 → B(2,5) Point C: From C, go down to x-axis: x=-2, go left to y-axis: y=2 → C(-2,2) Point D: From D, go up to x-axis: x=3, go left to y-axis: y=-1 → D(3,-1) Point E: From E, go up to x-axis: x=-2, go left to y-axis: y=-2 → E(-2,-2) Point F: From F, go up to x-axis: x=2, go left to y-axis: y=-3 → F(2,-3) Example 2: Points on Axes Points that lie exactly on an axis have one coordinate equal to zero. Key Facts: • Points on the x-axis have coordinates (x, 0) • Points on the y-axis have coordinates (0, y) • The origin is (0, 0) Examples: G(5, 0) lies on x-axis, 5 units right of origin H(0, -3) lies on y-axis, 3 units down from origin I(-4, 0) lies on x-axis, 4 units left of origin J(0, 2) lies on y-axis, 2 units up from origin Application Practice If a point lies on the x-axis and is 7 units to the left of the origin, what are its coordinates? If a point lies on the y-axis and is 4 units above the origin, what are its coordinates? A point is located at (-3, 0). On which axis does it lie? Describe how to find the coordinates of a point that's 2 units right and 5 units down from the origin. What are the coordinates of a point that lies exactly halfway between (0, 6) and (0, -2)? Distance Between Points We can calculate the distance between points that share either the same x-coordinate or the same y-coordinate by finding the difference between their coordinates. Calculating Distance: For points with same x-coordinate (vertical line): Distance = |y₂ - y₁| (absolute difference of y-coordinates) For points with same y-coordinate (horizontal line): Distance = |x₂ - x₁| (absolute difference of x-coordinates) Absolute value ensures distance is always positive. Example 1: Vertical Distance Find the distance between A(3, 2) and B(3, 7) Solution: Step 1: Notice both points have x-coordinate = 3 Step 2: Points lie on vertical line x = 3 Step 3: Distance = |y₂ - y₁| = |7 - 2| = |5| = 5 Step 4: The points are 5 units apart vertically Visual: A is at (3,2), B is at (3,7). To go from A to B, move 5 units up. Example 2: Horizontal Distance Find the distance between C(-4, 5) and D(2, 5) Solution: Step 1: Notice both points have y-coordinate = 5 Step 2: Points lie on horizontal line y = 5 Step 3: Distance = |x₂ - x₁| = |2 - (-4)| = |2 + 4| = |6| = 6 Step 4: The points are 6 units apart horizontally Visual: C is at (-4,5), D is at (2,5). To go from C to D, move 6 units right. Example 3: Distance Involving Negative Coordinates Find the distance between E(-3, -2) and F(-3, -7) Solution: Step 1: Both points have x-coordinate = -3 Step 2: Points lie on vertical line x = -3 Step 3: Distance = |y₂ - y₁| = |-7 - (-2)| = |-7 + 2| = |-5| = 5 Step 4: The points are 5 units apart vertically Note: Even with negative coordinates, distance is positive! Common Mistake: Incorrect: For points (-2, 3) and (-2, -4), calculating distance as 3 - (-4) = -1 Correct: Use absolute value: |3 - (-4)| = |3 + 4| = |7| = 7 Distance is always positive! Technique Practice Find the distance between (5, 2) and (5, 9). Find the distance between (-3, 4) and (2, 4). Points A and B have coordinates (-1, 5) and (-1, -3). What is the distance between them? Explain why distance is always positive. Two points have coordinates (x, 3) and (x, -2). What is the distance between them? Reflections Across Axes Reflection is a transformation that flips a point over a line (the axis). When reflecting across an axis, one coordinate changes sign while the other stays the same. Reflection Rules: Reflection across x-axis: (x, y) → (x, -y) • y-coordinate changes sign • x-coordinate stays the same Reflection across y-axis: (x, y) → (-x, y) • x-coordinate changes sign • y-coordinate stays the same Reflection across origin: (x, y) → (-x, -y) • Both coordinates change sign Example 1: Reflection Across x-axis Reflect point P(4, 3) across the x-axis. Solution: Original point: P(4, 3) Reflection across x-axis changes y to -y New point: P'(4, -3) Visual: P(4,3) is 3 units above x-axis. Its reflection P'(4,-3) is 3 units below x-axis. Example 2: Reflection Across y-axis Reflect point Q(-5, 2) across the y-axis. Solution: Original point: Q(-5, 2) Reflection across y-axis changes x to -x New point: Q'(5, 2) Visual: Q(-5,2) is 5 units left of y-axis. Its reflection Q'(5,2) is 5 units right of y-axis. Example 3: Multiple Reflections Point R(3, -4) is reflected across the x-axis, then that result is reflected across the y-axis. What are the final coordinates? Solution: Step 1: Reflect (3, -4) across x-axis → (3, 4) Step 2: Reflect (3, 4) across y-axis → (-3, 4) Final coordinates: (-3, 4) Note: This is equivalent to reflecting across the origin: (3, -4) → (-3, 4) Original Point Reflection Across Image Point Pattern (2, 5) x-axis (2, -5) y changes sign (-3, 4) x-axis (-3, -4) y changes sign (6, -1) y-axis (-6, -1) x changes sign (-2, -3) y-axis (2, -3) x changes sign (4, 2) origin (-4, -2) both change sign Method Practice Reflect point (5, 3) across the x-axis. Reflect point (-2, 4) across the y-axis. What are the coordinates of the reflection of (-3, -5) across the x-axis? Point A(4, -2) is reflected across the y-axis to get point B. What are B's coordinates? If reflecting point P across the y-axis gives (3, 5), what were P's original coordinates? Finding Missing Coordinates Sometimes we know certain properties about points (like they form a rectangle or lie on a vertical line) and need to find missing coordinates. This requires understanding geometric relationships on the coordinate plane. Strategies for Finding Missing Coordinates: 1. Points on vertical lines share the same x-coordinate 2. Points on horizontal lines share the same y-coordinate 3. Opposite vertices of rectangles have special relationships 4. Midpoints have coordinates that are averages of endpoints 5. Symmetric points have coordinates with specific sign patterns Example 1: Points on Vertical Line Points A(3, 5) and B(3, y) are 8 units apart. Find y. Solution: Step 1: Both points have x=3, so they're on vertical line Step 2: Distance = |y₂ - y₁| = |y - 5| = 8 Step 3: Two possibilities: y - 5 = 8 OR y - 5 = -8 Step 4: Solve: y = 13 OR y = -3 Step 5: Check: |13-5|=8 ✓, |-3-5|=|-8|=8 ✓ There are two possible points: B(3, 13) or B(3, -3) Example 2: Rectangle Vertices Three vertices of a rectangle are A(2, 1), B(2, 5), C(6, 5). Find D, the fourth vertex. Solution: Step 1: Plot points: A(2,1), B(2,5), C(6,5) Step 2: Notice AB is vertical (x=2 for both) Step 3: BC is horizontal (y=5 for both) Step 4: In rectangle, opposite sides are equal and parallel Step 5: D must have same x as C (6) and same y as A (1) Step 6: D = (6, 1) Check: AB and CD are vertical, BC and AD are horizontal. Example 3: Finding y-intercept A line passes through points (3, 7) and (3, -2). Where does it cross the y-axis? Solution: Step 1: Both points have x=3, so line is vertical: x = 3 Step 2: A vertical line x=3 never crosses y-axis! Step 3: Wait, check: y-axis is x=0, line is x=3 Step 4: These are parallel lines, they don't intersect Answer: The vertical line x=3 does not cross the y-axis. Verification Practice Points (4, y) and (4, -3) are 9 units apart. Find y. Three vertices of a square are (1, 2), (1, 6), and (5, 6). Find the fourth vertex. If point (x, 5) and point (2, 5) are 7 units apart, find x. A horizontal line contains point (-3, 4). What is the y-coordinate of every point on this line? Point A(2, 3) and point B(x, 3) are 10 units apart. Find both possible values for x. Real-World Applications Coordinate geometry has many practical applications in everyday life: maps, GPS navigation, video games, architecture, and more. Understanding coordinates helps us navigate and describe positions in the real world. Example 1: Map Coordinates A city map uses a coordinate system where each unit represents 1 km. The library is at (3, 4), the school is at (-2, 1), and the park is at (0, -3). Questions and Solutions: a) How far is the library from the school horizontally? Horizontal distance = |3 - (-2)| = |5| = 5 km b) How far is the park from the school vertically? Vertical distance = |-3 - 1| = |-4| = 4 km c) If you start at the origin and go to the library, then to the park, what's your total horizontal movement? Origin to library: 3 km right Library to park: |0 - 3| = 3 km left Total = 3 + 3 = 6 km horizontal movement Example 2: Chess Board Coordinates A chess board can be represented as an 8×8 coordinate grid with columns a-h and rows 1-8. Convert chess notation to coordinates: a1 = (1,1), h8 = (8,8). Problem: A knight at (4,4) moves in an "L" shape: 2 squares in one direction, then 1 square perpendicular. List possible coordinates after one move. Possible moves from (4,4): 1. (4+2, 4+1) = (6,5) 2. (4+2, 4-1) = (6,3) 3. (4-2, 4+1) = (2,5) 4. (4-2, 4-1) = (2,3) 5. (4+1, 4+2) = (5,6) 6. (4+1, 4-2) = (5,2) 7. (4-1, 4+2) = (3,6) 8. (4-1, 4-2) = (3,2) Example 3: Battleship Game In Battleship, players use coordinates to guess ship locations. A ship occupies points (3,2), (3,3), (3,4), (3,5). Analysis: • All points have x=3, so ship is vertical • Length = distance between endpoints = |5-2| = 3 units • Actually, 4 points: (3,2), (3,3), (3,4), (3,5) means length 4 • Midpoint = average of endpoints = (3, (2+5)/2) = (3, 3.5) If hit at (3,3): Still need hits at (3,2), (3,4), (3,5) to sink ship Application Practice A treasure is buried at (-5, 8) on a map. Starting from (0,0), describe how to get to the treasure. In a city grid, a hospital is at (2,4) and a fire station is at (2,-3). How far apart are they? A video game character starts at (0,0), moves to (5,2), then to (3,6). What's the net displacement from start? On a football field, a player runs from (10,20) to (10,40) to (30,40). How far did they run total? Create a real-world situation that uses coordinate geometry. Geometric Shapes on Coordinate Plane We can plot polygons on the coordinate plane and analyze their properties. This helps visualize geometric concepts and calculate measurements like perimeter and area. Example 1: Plotting a Rectangle Plot rectangle with vertices A(1,1), B(1,4), C(6,4), D(6,1). Analysis: Sides: AB: vertical, length = |4-1| = 3 units BC: horizontal, length = |6-1| = 5 units CD: vertical, length = |1-4| = 3 units DA: horizontal, length = |1-6| = 5 units Properties: • Opposite sides equal (rectangle property) • Perimeter = 3+5+3+5 = 16 units • Area = length × width = 5 × 3 = 15 square units Example 2: Right Triangle Plot right triangle with vertices P(0,0), Q(4,0), R(0,3). Analysis: Sides: PQ: horizontal, length = |4-0| = 4 units (along x-axis) PR: vertical, length = |3-0| = 3 units (along y-axis) QR: diagonal (hypotenuse) Right angle at P: because sides along axes are perpendicular Area: ½ × base × height = ½ × 4 × 3 = 6 square units Example 3: Symmetric Figure Plot points A(2,3), B(2,-3), C(-2,3), D(-2,-3). Analysis: Symmetries: • A and B are reflections across x-axis • A and C are reflections across y-axis • A and D are reflections across origin • All points form rectangle centered at origin Dimensions: Width = |2 - (-2)| = 4 units Height = |3 - (-3)| = 6 units Center = ((2+(-2))/2, (3+(-3))/2) = (0,0) Shape Vertices Key Property Perimeter/Area Clue Square (1,1),(1,4),(4,4),(4,1) All sides equal Side length = 3 Rectangle (2,1),(2,5),(7,5),(7,1) Opposite sides equal Length=5, Width=4 Right Triangle (0,0),(3,0),(0,4) Right angle at origin Legs: 3 and 4 Isosceles Triangle (0,0),(4,0),(2,3) Two equal sides Base=4, Height=3 Skills Practice Plot points (0,0), (0,5), (4,5), (4,0). What shape is formed? Calculate perimeter. A triangle has vertices at (1,1), (1,4), (5,1). Is it a right triangle? How do you know? Plot a square with side length 3 units, centered at the origin. A rectangle has vertices at (-2,1), (-2,4), (3,4). Find the fourth vertex. Create coordinates for an isosceles triangle with base on x-axis. Cumulative Exercises Plot and label these points: A(2,3), B(-4,1), C(0,-5), D(-3,-2) Identify the quadrant or axis for each point: (5,-2), (-3,-4), (0,6), (-7,0) Find the distance between (2,5) and (2,9). Reflect point (-4,3) across the y-axis. Points (x,2) and (5,2) are 8 units apart. Find x. A rectangle has vertices at (1,2), (1,6), (4,6). Find the fourth vertex. If a point is reflected across the x-axis to become (3,-5), what were its original coordinates? On a map, a school is at (-3,4) and a park is at (2,4). How far apart are they? Plot a right triangle with legs of length 3 and 4 units, with right angle at origin. Three vertices of a square are (-2,1), (-2,5), (2,5). Find the fourth vertex. Find the distance between (-5,0) and (3,0). Point P(4,-3) is reflected across the x-axis, then that result is reflected across the y-axis. Find final coordinates. A vertical line contains point (3,-2). What is the x-coordinate of every point on this line? Points A(0,0), B(0,6), C(8,6), D(8,0) form what shape? Calculate its area. Create a real-world problem using coordinate geometry and solve it. Show/Hide Answers Exercise 1: A(2,3): Quadrant I, B(-4,1): Quadrant II, C(0,-5): y-axis, D(-3,-2): Quadrant III Exercise 2: (5,-2): Quadrant IV, (-3,-4): Quadrant III, (0,6): y-axis, (-7,0): x-axis Exercise 3: Distance = |9-5| = 4 units (vertical distance) Exercise 4: Reflection across y-axis: (-4,3) → (4,3) Exercise 5: |x-5| = 8, so x-5=8 or x-5=-8. x=13 or x=-3 Exercise 6: Fourth vertex: (4,2) (completes rectangle) Exercise 7: Reflection across x-axis: (x,y)→(x,-y). Original point: (3,5) Exercise 8: Distance = |2-(-3)| = 5 units (horizontal distance) Exercise 9: Possible vertices: (0,0), (3,0), (0,4) OR (0,0), (4,0), (0,3) Exercise 10: Fourth vertex: (2,1) (completes square) Exercise 11: Distance = |3-(-5)| = 8 units (horizontal distance) Exercise 12: (4,-3) → x-axis reflection → (4,3) → y-axis reflection → (-4,3) Exercise 13: x-coordinate = 3 for all points on this vertical line Exercise 14: Rectangle with width=8, height=6. Area = 8×6 = 48 square units Exercise 15: Sample problem: On a city grid (1 unit = 1 block), library at (2,3), school at (2,-4). How many blocks between them? Solution: Both have x=2, so vertical distance = |3-(-4)| = 7 blocks Conclusion/Recap Excellent work! You've now mastered the fundamentals of coordinate geometry. You can plot points in all four quadrants, read coordinates from graphs, calculate distances, perform reflections, and apply these skills to real-world situations. Key Concepts to Remember: 1. Coordinate System: (x,y) = (horizontal, vertical) from origin 2. Quadrants: I(+,+), II(-,+), III(-,-), IV(+,-) 3. Plotting Points: Start at origin, move horizontally then vertically 4. Distance: For same x: |y₂-y₁|, for same y: |x₂-x₁| 5. Reflections: x-axis: (x,y)→(x,-y), y-axis: (x,y)→(-x,y) 6. Geometric Shapes: Use coordinates to plot and analyze polygons 7. Real Applications: Maps, navigation, games, and more use coordinates Common Mistakes to Avoid: • Switching x and y coordinates • Forgetting negative signs when plotting • Calculating distance without absolute value • Confusing reflection rules • Not starting at origin when plotting Coordinate geometry is the foundation for more advanced mathematics and has countless applications in science, engineering, computer graphics, and everyday life. Keep practicing by looking for coordinate systems around you—in maps, video games, or even seating charts! Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c
Get a Free Gold Subscription Upgrade when you join.