Overview
This chapter introduces the Cartesian (coordinate) plane and the language of coordinate geometry. Students learn how two perpendicular number lines (x-axis and y-axis) form a plane with an origin and four quadrants, how to represent locations by ordered pairs (x, y), and how to plot and read points. The chapter builds basic spatial reasoning and provides foundational tools for graphing, analytic geometry and connecting algebra with geometry.
Learning Objectives
- Define the Cartesian coordinate system and state the role of x-axis, y-axis and the origin
- Explain the concept of coordinates of a point and the procedure to locate a point in all four quadrants
- Plot given points on the coordinate plane and interpret their positions relative to the axes and origin
- Derive the distance formula between two points using the Pythagorean theorem
- Apply the distance formula to calculate the distance between any two given points in the plane
- Derive the section formula for internal division of a line segment in a given ratio
- Use the section formula to find coordinates of a point dividing a segment internally (including the midpoint as a special case)
- Calculate the coordinates of the midpoint of a line segment using the midpoint formula
Topics in this chapter
8 topics · tap a topic title to jump straight to it.
Cartesian Coordinate System
Cartesian Coordinate System
Key Point: Distance between two points P(x1, y1) and Q(x2, y2): d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
The Cartesian Coordinate System (or rectangular coordinate system) is a two-dimensional plane used to specify the position of points. It consists of two perpendicular number lines that intersect at a point called the origin.
Key parts:
- X-axis: the horizontal number line.
- Y-axis: the vertical number line.
- Origin (O): the point where the axes cross, labeled (0, 0).
- Coordinates / Ordered pair: every point P in the plane is represented by an ordered pair (x, y), where x is the abscissa (distance along the x-axis) and y is the ordinate (distance along the y-axis).
How to locate a point (x, y): start at the origin, move x units along the x-axis (to the right if x > 0, to the left if x < 0), then move y units parallel to the y-axis (up if y > 0, down if y < 0). Mark that location and label it (x, y).
Quadrants: the axes divide the plane into four regions called quadrants numbered I to IV (counterclockwise):
- Quadrant I: x > 0, y > 0 (both positive)
- Quadrant II: x < 0, y > 0
- Quadrant III: x < 0, y < 0
- Quadrant IV: x > 0, y < 0
Uses and extensions: the Cartesian system allows plotting points, drawing graphs of equations, measuring distances between points (using the distance formula derived from the Pythagorean theorem), finding midpoints, and describing straight lines and shapes precisely.
- Plot the point (3, 2): from origin move 3 units right, then 2 units up and mark the point.
- Find where (-4, 1) lies: from origin move 4 units left, then 1 unit up — this point is in Quadrant II.
- City grid: streets labeled as east–west (x) and north–south (y); an address (5, 7) can represent 5 blocks east and 7 blocks north of a reference intersection.
- Computer graphics: on a screen, objects are placed using (x, y) coordinates to control horizontal and vertical positions.
- Robot navigation: a robot on a factory floor uses coordinates to move to specific (x, y) target locations.
- \[Distance between two points P(x1\]\[y1) and Q(x2\]\[y2): d = sqrt((x2 - x1)^2 + (y2 - y1)^2).\]
- \[Midpoint of segment PQ: M = ((x1 + x2)/2\]\[(y1 + y2)/2).\]
- \[Slope (gradient) of the line through P and Q: m = (y2 - y1)/(x2 - x1) (useful for lines\]\[undefined if x1 = x2).\]
- \[Equation of horizontal line through y = k and vertical line through x = h.\]
Coordinates of a Point
Coordinates of a Point
Key Point: Coordinates of a point: (x, y) where x = abscissa, y = ordinate.
What are coordinates? In the Cartesian (coordinate) plane, every point is identified by an ordered pair of numbers (x, y) called its coordinates. The first number x is the abscissa (horizontal position) and the second number y is the ordinate (vertical position).
The Cartesian plane: Two number lines that meet at right angles form the plane: the horizontal line is the x-axis and the vertical line is the y-axis. Their intersection is the origin O with coordinates (0, 0). The plane is divided into four quadrants:
- Quadrant I: x > 0, y > 0
- Quadrant II: x < 0, y > 0
- Quadrant III: x < 0, y < 0
- Quadrant IV: x > 0, y < 0
How to read and plot an ordered pair (x, y):
- Start at the origin (0, 0).
- Move |x| units along the x-axis: right if x > 0, left if x < 0.
- From that point, move |y| units parallel to the y-axis: up if y > 0, down if y < 0.
- Mark the point and label it with (x, y).
Special cases: Points with x = 0 lie on the y-axis (e.g., (0, 5)). Points with y = 0 lie on the x-axis (e.g., (4, 0)).
Useful observations: If two points have the same x-coordinate, they lie on a vertical line. If they have the same y-coordinate, they lie on a horizontal line.
Why it matters (intuition and uses): Coordinates let us convert geometric problems into algebraic ones. They are used in maps, navigation, computer graphics, robotics, architecture and everyday grid systems (city streets, seating charts, spreadsheet cell addresses).
- Plot the point (3, 2): From origin move 3 units right, then 2 units up. The point lies in Quadrant I.
- Plot the point (-4, 1): From origin move 4 units left, then 1 unit up. The point lies in Quadrant II.
- Point (0, -3) lies on the y-axis 3 units below the origin because x = 0.
- Distance of (3, 4) from the origin: sqrt(3^2 + 4^2) = 5. (This uses the distance-from-origin formula.)
- Midpoint of A(2, 3) and B(4, -1): ((2+4)/2, (3+(-1))/2) = (3, 1). Plotting these shows the midpoint lies midway on the segment AB.
- \[Coordinates of a point: (x\]\[y) where x = abscissa\]\[y = ordinate.\]
- \[Distance of point (x\]\[y) from origin O(0, 0): d = sqrt(x^2 + y^2).\]
- \[Distance between two points P(x1\]\[y1) and Q(x2\]\[y2): d = sqrt((x2 - x1)^2 + (y2 - y1)^2).\]
- \[Midpoint of segment joining P(x1\]\[y1) and Q(x2\]\[y2): M = ((x1 + x2)/2\]\[(y1 + y2)/2).\]
Plotting Points
Plotting Points
Key Point: Point coordinate notation: P(x, y) — x is horizontal coordinate, y is vertical coordinate.
What is plotting points? Plotting points means placing points on the Cartesian plane (or coordinate plane) using ordered pairs (x, y). The Cartesian plane is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. Their intersection is the origin, O(0, 0).
Basic terms:
- Ordered pair (x, y): x is the horizontal coordinate (move right if x > 0, left if x < 0). y is the vertical coordinate (move up if y > 0, down if y < 0).
- Quadrants: The plane is divided into four quadrants. Signs of coordinates in each quadrant are: I (+,+), II (-,+), III (-,-), IV (+,-).
- Points on axes: If y = 0 the point lies on the x-axis (e.g., (3, 0)). If x = 0 the point lies on the y-axis (e.g., (0, -2)).
Step-by-step procedure to plot a point (x, y):
- Draw the x- and y-axes and mark equal units (choose a suitable scale).
- Start at the origin O(0, 0).
- Move horizontally to x: right if x > 0, left if x < 0.
- From that position, move vertically to y: up if y > 0, down if y < 0.
- Mark the point and label it with its ordered pair.
Common mistakes to avoid: Mixing the order of coordinates (x, y) — always x first, then y; inconsistent scale on axes; forgetting sign of coordinates when moving left/down.
Why it matters (intuitively and practically): Plotting points gives a visual way to represent locations, positions, relationships and shapes. It is the foundation for graphing lines, curves and geometric figures and is used in maps, game grids, engineering drawings and data plots.
- Plot (3, 2): From origin move 3 units right, then 2 units up; mark and label (3, 2).
- Plot (-4, 1): From origin move 4 units left, then 1 unit up; mark and label (-4, 1) (lies in Quadrant II).
- Plot (0, -3): x = 0 so stay on y-axis; move 3 units down from origin; mark (0, -3) on the negative y-axis.
- Plot points (2, 1), (2, 4), (5, 1) and (5, 4) and join to form a rectangle. Use this to verify coordinates of its vertices and sides parallel to axes.
- Real-life: Represent seating in a classroom as coordinates where x = column number and y = row number. Seat (3, 2) means column 3, row 2.
- \[Point coordinate notation: P(x\]\[y) — x is horizontal coordinate\]\[y is vertical coordinate.\]
- \[Distance between two points P1(x1\]\[y1) and P2(x2\]\[y2): d = sqrt((x2 - x1)^2 + (y2 - y1)^2).\]
- \[Midpoint of P1(x1\]\[y1) and P2(x2\]\[y2): M = ((x1 + x2)/2\]\[(y1 + y2)/2).\]
- \[Slope (gradient) of line through P1 and P2: m = (y2 - y1) / (x2 - x1) (x2 ≠ x1).\]
- \[Section formula (internal division in ratio m:n of segment joining P1 and P2): ((m x2 + n x1)/(m + n)\]\[(m y2 + n y1)/(m + n)).\]
Quadrants and Signs of Coordinates
Quadrants and Signs of Coordinates
Key Point: Sign rules for quadrants: (x, y) in Quadrant I if x>0, y>0; Quadrant II if x<0, y>0; Quadrant III if x<0, y<0; Quadrant IV if x>0, y<0.
The Cartesian plane (coordinate plane) is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. Their intersection is the origin O with coordinates (0, 0). Every point in the plane is represented by an ordered pair (x, y), where x is the abscissa (horizontal distance from the origin) and y is the ordinate (vertical distance from the origin).
The axes divide the plane into four regions called quadrants. By convention they are numbered counterclockwise starting from the upper-right region:
- Quadrant I: x > 0 and y > 0 (both positive)
- Quadrant II: x < 0 and y > 0 (x negative, y positive)
- Quadrant III: x < 0 and y < 0 (both negative)
- Quadrant IV: x > 0 and y < 0 (x positive, y negative)
Points that lie on the x-axis have coordinates of the form (x, 0); points on the y-axis have coordinates (0, y). The origin (0, 0) lies on both axes. To determine the quadrant of a point, look at the signs of its coordinates: sign(+,+) → I, sign(-,+) → II, sign(-,-) → III, sign(+,-) → IV.
Understanding quadrants helps in plotting points, reading graphs, and solving geometry problems in coordinate form. Note: some computer graphics and screen coordinate systems use the top-left as origin with y increasing downward; that is different from the standard mathematical convention where y increases upward.
- Point A(3, 4): x = 3 > 0 and y = 4 > 0 → Quadrant I.
- Point B(-5, 2): x = -5 < 0 and y = 2 > 0 → Quadrant II.
- Point C(-2, -6): x = -2 < 0 and y = -6 < 0 → Quadrant III.
- Point D(7, -3): x = 7 > 0 and y = -3 < 0 → Quadrant IV.
- Point E(4, 0) lies on the x-axis (not in any quadrant).
- Point F(0, -3) lies on the y-axis (not in any quadrant).
- \[Sign rules for quadrants: (x\]\[y) in Quadrant I if x>0\]\[y>0\]\[Quadrant II if x<0\]\[y>0\]\[Quadrant III if x<0\]\[y<0\]\[Quadrant IV if x>0\]\[y<0.\]
- \[Point on axes: x-axis → (x, 0)\]\[y-axis → (0\]\[y)\]\[origin → (0, 0).\]
- \[Distance of point (x\]\[y) from origin O: sqrt(x^2 + y^2).\]
- \[Distance between two points P(x1\]\[y1) and Q(x2\]\[y2): sqrt((x2 - x1)^2 + (y2 - y1)^2).\]
- \[Midpoint of PQ: ((x1 + x2)/2\]\[(y1 + y2)/2).\]
Distance between Two Points
Distance between Two Points
Key Point: Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
What it is: The distance between two points in the coordinate plane is the length of the straight line segment joining them. If the points are A(x1, y1) and B(x2, y2), we find this length using the Pythagorean theorem.
Derivation (using Pythagoras): Draw a right triangle by dropping a horizontal and a vertical from A and B. The horizontal leg has length |x2 − x1| and the vertical leg has length |y2 − y1|. By Pythagoras, the distance d = AB satisfies
d = sqrt((x2 − x1)2 + (y2 − y1)2)
Notes: signs of differences do not matter because of squaring; distance is always non‑negative. Special cases: if y1 = y2 (same horizontal line) then d = |x2 − x1|. If x1 = x2 (same vertical line) then d = |y2 − y1|.
- Example 1 — Simple positive coordinates: A(2, 3), B(5, 7). Compute Δx = 5 − 2 = 3, Δy = 7 − 3 = 4. Distance = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.
- Example 2 — With negative coordinates: A(−1, 4), B(3, −2). Δx = 3 − (−1) = 4, Δy = −2 − 4 = −6. Distance = sqrt(4^2 + (−6)^2) = sqrt(16 + 36) = sqrt(52) = 2√13 (approx. 7.211).
- Example 3 — Horizontal/vertical cases: (a) A(2, 5), B(8, 5): y's equal so distance = |8 − 2| = 6. (b) A(4, 1), B(4, −3): x's equal so distance = |1 − (−3)| = 4.
- \[Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)\]
- \[Squared distance (useful to avoid square root): d^2 = (x2 - x1)^2 + (y2 - y1)^2\]
- \[Horizontal points (y1 = y2): d = |x2 - x1|\]
- \[Vertical points (x1 = x2): d = |y2 - y1|\]
Midpoint of a Line Segment
Midpoint of a Line Segment
Key Point: Midpoint formula (2D): M\u2190\u00A0((x1 + x2)/2, (y1 + y2)/2)
Definition: The midpoint of a line segment AB in the coordinate plane is the point M that is exactly halfway between A and B. It divides the segment into two equal parts so that MA = MB.
Derivation / Reasoning: Let A(x1, y1) and B(x2, y2). The horizontal distance from A to M must equal the horizontal distance from M to B, and similarly for the vertical distances. Thus if M=(x, y) we have x - x1 = x2 - x and y - y1 = y2 - y. Solving gives x = (x1 + x2)/2 and y = (y1 + y2)/2. This shows the midpoint coordinates are the averages of the corresponding coordinates of the endpoints.
Key properties:
- Midpoint M is the point that divides AB in the ratio 1:1 (internal division).
- M is equidistant from A and B: MA = MB (you can check using the distance formula).
- The midpoint of the diagonals of a rectangle (or parallelogram) is the same point — the centre.
Usage: The midpoint formula is used in geometry problems, construction, map and navigation tasks, computer graphics and animations where you need the centre between two points.
- Example 1: Find the midpoint of A(2, 3) and B(6, 7). Solution: M = ((2+6)/2, (3+7)/2) = (4, 5).
- Example 2: Midpoint of A(-1, 4) and B(5, -2). Solution: M = ((-1+5)/2, (4+(-2))/2) = (4/2, 2/2) = (2, 1).
- Example 3 (horizontal segment): A(0, 0) and B(4, 0). Midpoint M = ((0+4)/2, (0+0)/2) = (2, 0) — centre on the x-axis.
- Example 4 (vertical segment): A(3, 2) and B(3, 8). Midpoint M = ((3+3)/2, (2+8)/2) = (3, 5) — same x-coordinate, midpoint of y-values.
- \[Midpoint formula (2D): M\u2190\u00A0((x1 + x2)/2\]\[(y1 + y2)/2)\]
- \[Midpoint as section in ratio 1:1: M = ((1*x2 + 1*x1)/(1+1)\]\[(1*y2 + 1*y1)/(1+1))\]
- \[Property: MA = MB\]\[check using distance formula: sqrt((x - x1)^2 + (y - y1)^2) = sqrt((x - x2)^2 + (y - y2)^2)\]
- \[Special cases: horizontal segment (y1 = y2) -> M_x = (x1 + x2)/2\]\[y = y1\]\[vertical segment (x1 = x2) -> M_y = (y1 + y2)/2\]\[x = x1\]
Section Formula (Division of a Line Segment)
Section Formula (Division of a Line Segment)
Key Point: Internal division (AP : PB = m : n): P = ( (n·x1 + m·x2)/(m + n) , (n·y1 + m·y2)/(m + n) )
What it is: The section formula gives the coordinates of a point P that divides a line segment AB joining A(x1, y1) and B(x2, y2) in a given ratio.
Internal division: If P divides AB internally in the ratio AP : PB = m : n, then P is a weighted average of A and B. Using proportional distances,
x-coordinate of P = x1 + (m/(m+n))(x2 - x1) = (n·x1 + m·x2)/(m + n)
y-coordinate of P = y1 + (m/(m+n))(y2 - y1) = (n·y1 + m·y2)/(m + n)
This formula can be derived by writing P = A + (m/(m+n))(B - A) or by using similar triangles (proportional segments on coordinate axes).
Midpoint (special case): If m = n = 1 (i.e., AP = PB), P is the midpoint. Coordinates: ((x1 + x2)/2, (y1 + y2)/2).
External division: If P divides AB externally in the ratio AP : PB = m : n (P lies on the line AB but outside the segment AB), the coordinates are
P = ( (m·x2 − n·x1)/(m − n), (m·y2 − n·y1)/(m − n) ).
Interpretation: Think of the coordinates of P as a weighted average of the endpoints, where weights are proportional to the opposite segment lengths. The section formula is widely used in geometry, physics (centroid, center of mass for two-point systems), navigation and mapping (interpolating positions on straight paths).
- Example 1 (internal division): A(2, 3) and B(8, 7). Find point P that divides AB in the ratio 2:3 (AP:PB = 2:3). Using formula P = ((n·x1 + m·x2)/(m+n), (n·y1 + m·y2)/(m+n)) with m=2, n=3 gives x = (3·2 + 2·8)/(2+3) = (6 + 16)/5 = 22/5 = 4.4, y = (3·3 + 2·7)/5 = (9 + 14)/5 = 23/5 = 4.6. So P(4.4, 4.6).
- Example 2 (midpoint): A(−1, 4), B(5, −2). Midpoint M = ((−1+5)/2, (4+(−2))/2) = (2, 1).
- Example 3 (external division): A(1, 2), B(7, 8). Find point P that divides AB externally in the ratio 2:1 (AP:PB = 2:1). Using external formula P = ((m·x2 − n·x1)/(m − n), (m·y2 − n·y1)/(m − n)) with m=2, n=1 gives x = (2·7 − 1·1)/(2−1) = (14 − 1)/1 = 13, y = (2·8 − 1·2)/(2−1) = (16 − 2)/1 = 14. So P(13, 14) which lies on the line AB extended beyond B.
- Example 4 (application): Suppose two charging stations at A(0,0) and B(10,0). A small town should be located such that its distance from A and B are in ratio 1:3 (closer to A). Coordinates: P = ((3·0 + 1·10)/(1+3), (3·0 + 1·0)/4) = (10/4, 0) = (2.5, 0).
- \[Internal division (AP : PB = m : n): P = ( (n·x1 + m·x2)/(m + n)\]\[(n·y1 + m·y2)/(m + n) )\]
- \[Midpoint (m = n = 1): M = ( (x1 + x2)/2\]\[(y1 + y2)/2 )\]
- \[External division (AP : PB = m : n externally): P = ( (m·x2 − n·x1)/(m − n)\]\[(m·y2 − n·y1)/(m − n) )\]
- \[Vector form: If P divides AB in ratio m:n then →OP = (n/(m+n))→OA + (m/(m+n))→OB\]
Problems and Applications in Coordinate Geometry
Problems and Applications in Coordinate Geometry
Key Point: Coordinates of a point: (x, y)
Overview: Coordinate geometry (Cartesian geometry) places points on a plane using ordered pairs (x, y). The horizontal axis is the x-axis and the vertical axis is the y-axis. Every point has an abscissa (x) and an ordinate (y). The plane is divided into four quadrants.
Key ideas and how they arise:
- Plotting points: To plot (x, y), move x units along x-axis (right for +, left for −) and y units along y-axis (up for +, down for −).
- Distance formula: The distance between A(x1, y1) and B(x2, y2) is found by making a right triangle and using Pythagoras: d = sqrt((x2 − x1)^2 + (y2 − y1)^2).
- Midpoint and section formula: Midpoint of AB is ((x1 + x2)/2, (y1 + y2)/2). More generally, a point P dividing AB internally in ratio m:n (AP:PB = m:n) has coordinates P = ((n*x1 + m*x2)/(m+n), (n*y1 + m*y2)/(m+n)).
- Slope (gradient) of a line: Slope m of line through A(x1, y1) and B(x2, y2) is m = (y2 − y1)/(x2 − x1) (if x2 ≠ x1). Slope measures steepness and sign indicates rise/fall.
- Equation of a line: Using slope-point form: y − y1 = m(x − x1). Other common forms: slope-intercept y = mx + c, and general ax + by + c = 0.
- Area of triangle by coordinates: For vertices (x1,y1),(x2,y2),(x3,y3): area = (1/2) | x1(y2−y3) + x2(y3−y1) + x3(y1−y2) |.
Why these are useful: These formulae let you reduce geometric problems (lengths, midpoints, ratios, areas, straightness) to algebraic computation. That makes many geometry problems easier and allows application to real-life problems like distances on maps, construction plans, computer graphics, and navigation.
- 1) Distance between A(2, 3) and B(7, −1). Solution: d = sqrt((7−2)^2 + (−1−3)^2) = sqrt(25 + 16) = sqrt(41) ≈ 6.403.
- 2) Midpoint of A(−3, 4) and B(5, 10). Solution: M = ((−3+5)/2, (4+10)/2) = (1, 7).
- 3) Point dividing AB in ratio 2:1 (AP:PB = 2:1) for A(1, 2) and B(7, 8). Solution: P = ((n*x1 + m*x2)/(m+n), (n*y1 + m*y2)/(m+n)) with m=2, n=1 → P = ((1*1 + 2*7)/3, (1*2 + 2*8)/3) = (5, 6).
- 4) Slope and equation of line through (2, 3) and (5, 11). Slope m = (11−3)/(5−2) = 8/3. Equation: y − 3 = (8/3)(x − 2) or y = (8/3)x − 7/3.
- 5) Area of triangle with vertices (0,0), (4,0), (2,3). Area = 1/2 * |0*(0−3) + 4*(3−0) + 2*(0−0)| = 1/2 * 12 = 6 sq. units.
- \[Coordinates of a point: (x\]\[y)\]
- \[Distance between two points A(x1\]\[y1) and B(x2\]\[y2): d = sqrt((x2 − x1)^2 + (y2 − y1)^2)\]
- \[Midpoint of AB: M = ((x1 + x2)/2\]\[(y1 + y2)/2)\]
- \[Section formula (internal division): If P divides AB in ratio m:n (AP:PB = m:n)\]\[P = ((n*x1 + m*x2)/(m+n)\]\[(n*y1 + m*y2)/(m+n))\]
- \[Slope of line through A and B: m = (y2 − y1)/(x2 − x1) (x2 ≠ x1)\]
- \[Equation of a line (point-slope): y − y1 = m(x − x1)\]\[slope-intercept: y = mx + c\]
Key Concepts
- Coordinate Geometry
- Branch of geometry that uses a coordinate system (usually Cartesian) to represent and study geometric figures algebraically.
- Cartesian Plane
- A two-dimensional plane formed by two perpendicular number lines called the x-axis and y-axis.
- Origin
- The point where the x-axis and y-axis meet; coordinates (0,0).
- Axes (x-axis and y-axis)
- The two perpendicular reference lines in the Cartesian plane: horizontal is x-axis, vertical is y-axis.
- Point (Ordered Pair)
- A location in the plane represented by an ordered pair (x,y), where x is abscissa and y is ordinate.
- Abscissa
- The x-coordinate of a point in the plane (the horizontal component).
- Ordinate
- The y-coordinate of a point in the plane (the vertical component).
- Quadrants
- The four regions of the Cartesian plane separated by the axes: I (+,+), II (−,+), III (−,−), IV (+,−).
- Distance Formula
- Distance between (x1,y1) and (x2,y2): sqrt((x2−x1)^2 + (y2−y1)^2).
- Midpoint Formula
- Midpoint of segment joining (x1,y1) and (x2,y2) is ((x1+x2)/2, (y1+y2)/2).
- Section Formula (Internal Division)
- Coordinates of point dividing segment joining (x1,y1) and (x2,y2) internally in ratio m:n are ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)).
- Section Formula (External Division)
- Coordinates of point dividing segment externally in ratio m:n are ((mx2 − nx1)/(m−n), (my2 − ny1)/(m−n)), with m ≠ n.
- Slope (Gradient)
- Measure of steepness of a line through (x1,y1) and (x2,y2): m = (y2−y1)/(x2−x1), provided x2 ≠ x1.
- Equation of a Line
- Algebraic representation of a line; common form: y = mx + c where m is slope and c is y-intercept.
- x-intercept
- Point where a curve or line meets the x-axis (y = 0); written as (a,0).
- y-intercept
- Point where a curve or line meets the y-axis (x = 0); written as (0,b).
- Collinear Points
- Points that lie on the same straight line. Three points are collinear if area of triangle formed by them is zero.
- Locus
- The set of all points satisfying a given condition or rule.
- Area of Triangle (Using Coordinates)
- Area of triangle with vertices (x1,y1),(x2,y2),(x3,y3) is 1/2 |x1(y2−y3)+x2(y3−y1)+x3(y1−y2)|.
- Parallel and Perpendicular Lines (Slope Tests)
- Two lines are parallel if their slopes are equal; perpendicular if product of slopes = −1 (neither vertical).
Practice Questions
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In which quadrant does the point (−3, 5) lie? (a) Quadrant I (b) Quadrant II (c) Quadrant III (d) Quadrant IV / बिंदु (−3, 5) किस चतुर्थांश में स्थित है? (a) चतुर्थांश I (b) चतुर्थांश II (c) चतुर्थांश III (d) चतुर्थांश IV
Show answer
(b) Quadrant II / (b) चतुर्थांश II — x < 0 और y > 0 होने पर बिंदु चतुर्थांश II में होता है। / Since x = −3 < 0 and y = 5 > 0, the point lies in Quadrant II.
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What are the coordinates of a point that lies on the y-axis at 4 units above the origin? (a) (4, 0) (b) (0, 4) (c) (4, 4) (d) (−4, 0) / मूल बिंदु से 4 इकाई ऊपर y-अक्ष पर स्थित बिंदु के निर्देशांक क्या हैं? (a) (4, 0) (b) (0, 4) (c) (4, 4) (d) (−4, 0)
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(b) (0, 4) / (b) (0, 4) — y-अक्ष पर स्थित बिंदु का x-निर्देशांक 0 होता है। / A point on the y-axis has x-coordinate 0; 4 units above the origin gives (0, 4).
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What is the distance between the points A(2, 3) and B(5, 7)? (a) 3 (b) 4 (c) 5 (d) 7 / बिंदुओं A(2, 3) और B(5, 7) के बीच की दूरी क्या है? (a) 3 (b) 4 (c) 5 (d) 7
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(c) 5 / (c) 5 — d = √((5−2)² + (7−3)²) = √(9+16) = √25 = 5. / Using the distance formula: d = √((5−2)² + (7−3)²) = √(9+16) = 5.
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Find the midpoint of the segment joining A(−1, 4) and B(5, −2). / A(−1, 4) और B(5, −2) को मिलाने वाले रेखाखंड का मध्य बिंदु ज्ञात करें।
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Midpoint = (2, 1) / मध्य बिंदु = (2, 1) — M = ((−1+5)/2, (4−2)/2) = (4/2, 2/2) = (2, 1). Using the midpoint formula M = ((x₁+x₂)/2, (y₁+y₂)/2).
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The x-coordinate of a point is called its ______ and the y-coordinate is called its ______. / किसी बिंदु के x-निर्देशांक को ______ और y-निर्देशांक को ______ कहते हैं।
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Abscissa; Ordinate / भुज; कोटि — x-निर्देशांक को भुज (abscissa) और y-निर्देशांक को कोटि (ordinate) कहते हैं। / The x-coordinate is the abscissa (horizontal distance) and the y-coordinate is the ordinate (vertical distance).
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True or False: The point (0, 0) lies in Quadrant I. / सत्य या असत्य: बिंदु (0, 0) चतुर्थांश I में स्थित है।
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False / असत्य — (0, 0) मूल बिंदु (origin) है और यह किसी भी चतुर्थांश में नहीं, बल्कि दोनों अक्षों के प्रतिच्छेदन पर स्थित है। / The origin (0, 0) lies on both axes, not in any quadrant.
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A point P divides the segment joining A(1, 2) and B(7, 8) in the ratio 2:1 internally. Find the coordinates of P. / बिंदु P, A(1, 2) और B(7, 8) को मिलाने वाले रेखाखंड को 2:1 के अनुपात में अंतः विभाजित करता है। P के निर्देशांक ज्ञात करें।
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P = (5, 6) / P = (5, 6) — Section formula: P = ((2×7 + 1×1)/(2+1), (2×8 + 1×2)/(2+1)) = (15/3, 18/3) = (5, 6). / Using the section formula for internal division in ratio m:n: P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).
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What is the area of a triangle with vertices O(0,0), A(4,0), and B(2,3)? / शीर्षों O(0,0), A(4,0) और B(2,3) वाले त्रिभुज का क्षेत्रफल क्या है?
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Area = 6 square units / क्षेत्रफल = 6 वर्ग इकाई — Area = ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)| = ½|0(0−3)+4(3−0)+2(0−0)| = ½×12 = 6. / Using the coordinate area formula for a triangle with vertices at given coordinates.
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