Key Concepts
1. Cartesian System
The Cartesian system uses two perpendicular axes:
- The horizontal axis (xx-axis).
- The vertical axis (yy-axis).
Origin (OO): Intersection point of the axes (0,0)(0, 0).
Points are represented as ordered pairs (x,y)(x, y), where:
- xx is the abscissa (horizontal distance from the origin).
- yy is the ordinate (vertical distance from the origin).
2. Quadrants
The plane is divided into four quadrants:
- Quadrant I: x>0,y>0x > 0, y > 0.
- Quadrant II: x<0,y>0x < 0, y > 0.
- Quadrant III: x<0,y<0x < 0, y < 0.
- Quadrant IV: x>0,y<0x > 0, y < 0.
3. Plotting Points
- Start at the origin.
- Move horizontally to the xx-coordinate.
- Move vertically to the yy-coordinate.
Example: Plot (2,3)(2, 3).
- Move 2 units right (positive xx).
- Move 3 units up (positive yy).
4. Distance Formula
The distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is: d=(x2−x1)2+(y2−y1)2.d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}.
Example: Find the distance between A(1,2)A(1, 2) and B(4,6)B(4, 6). d=(4−1)2+(6−2)2=32+42=9+16=25=5.d = \sqrt{(4 – 1)^2 + (6 – 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.
5. Section Formula
A point P(x,y)P(x, y) dividing a line segment joining (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in the ratio m:nm:n is: P(x,y)=(mx2+nx1m+n,my2+ny1m+n).P\left(x, y\right) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right).
Example: Find the point dividing the line joining A(1,2)A(1, 2) and B(4,6)B(4, 6) in the ratio 2:12:1. x=2(4)+1(1)2+1=8+13=3.x = \frac{2(4) + 1(1)}{2 + 1} = \frac{8 + 1}{3} = 3. y=2(6)+1(2)2+1=12+23=4.67.y = \frac{2(6) + 1(2)}{2 + 1} = \frac{12 + 2}{3} = 4.67.
Point P=(3,4.67)P = (3, 4.67).
6. Midpoint Formula
The midpoint of a line segment joining (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is: M(x,y)=(x1+x22,y1+y22).M\left(x, y\right) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).
Example: Find the midpoint of A(1,2)A(1, 2) and B(4,6)B(4, 6). x=1+42=2.5, y=2+62=4.x = \frac{1 + 4}{2} = 2.5, \, y = \frac{2 + 6}{2} = 4.
Midpoint M=(2.5,4)M = (2.5, 4).
7. Area of a Triangle
The area of a triangle with vertices (x1,y1)(x_1, y_1), (x2,y2)(x_2, y_2), and (x3,y3)(x_3, y_3) is: Area=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣.\text{Area} = \frac{1}{2} \left| x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2) \right|.
Example: Find the area of a triangle with vertices A(1,2)A(1, 2), B(4,6)B(4, 6), and C(3,5)C(3, 5). Area=12∣1(6−5)+4(5−2)+3(2−6)∣.\text{Area} = \frac{1}{2} \left| 1(6 – 5) + 4(5 – 2) + 3(2 – 6) \right|. =12∣1(1)+4(3)+3(−4)∣.= \frac{1}{2} \left| 1(1) + 4(3) + 3(-4) \right|. =12∣1+12−12∣=12∣1∣=0.5.= \frac{1}{2} \left| 1 + 12 – 12 \right| = \frac{1}{2} \left| 1 \right| = 0.5.
Answer: Area = 0.5 square units0.5 \, \text{square units}.
Critical Analysis
- Conceptual Clarity:
- Understanding quadrants is essential for correctly identifying the position of points.
- Mastery of formulas like distance, section, and midpoint helps solve geometry problems efficiently.
- Applications:
- Coordinate geometry has real-world applications in navigation, engineering, and computer graphics.
- It lays the foundation for advanced topics like vectors and 3D geometry.
- Common Mistakes:
- Confusing the signs of coordinates in different quadrants.
- Misapplying formulas, especially in the section formula with incorrect ratios.
- Visualization:
- Drawing rough sketches improves accuracy.
- Graphical representation helps in better understanding spatial relationships.
Applications of Coordinate Geometry
1. Map Reading
- Identifying locations on a map using coordinates.
2. Navigation
- GPS systems use coordinate geometry to calculate routes and distances.
3. Design and Animation
- Used in computer-aided design (CAD) and animation software to position objects.
4. Physics
- Analyzing motion in two dimensions.
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