Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Quick Summary
Coordinate Geometry places geometric figures on the Cartesian plane and uses algebraic methods to find distances, midpoints, and points that divide segments in given ratios. This chapter provides the bridge between algebra and geometry. Numerical problems are the most common question type.
Key Concepts
- Cartesian Plane: Two perpendicular number lines (x-axis and y-axis) meeting at the origin O(0, 0).
- Quadrants: The plane is divided into four quadrants: I (+,+), II (−,+), III (−,−), IV (+,−).
- Collinear Points: Three or more points that lie on the same straight line.
Important Formulae
| Formula | Expression |
|---|---|
| Distance formula | d = √[(x₂−x₁)² + (y₂−y₁)²] |
| Distance from origin | d = √(x² + y²) |
| Midpoint formula | M = ((x₁+x₂)/2, (y₁+y₂)/2) |
| Section formula (internal) | P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)) |
| Centroid of triangle | G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) |
Special Cases
- Midpoint: Section formula with ratio 1:1 gives the midpoint.
- Trisection: A point divides a segment in ratio 1:2 or 2:1.
- Collinearity test: Three points A, B, C are collinear if AB + BC = AC (using distance formula).
Plotting Points
- To plot (−3, 4): Move 3 units left on x-axis, then 4 units up.
- Points on x-axis have y = 0.
- Points on y-axis have x = 0.
Important Questions
One Mark
1. Find the distance between (0, 0) and (3, 4). 2. What is the midpoint of (2, 6) and (4, 8)? 3. In which quadrant does the point (−3, −5) lie? 4. State the section formula for internal division. 5. Find the centroid of a triangle with vertices (1,1), (3,5), (5,3).
Short Answer — 3 marks
1. Show that the points A(1, 2), B(4, 6), C(−3, −2) form an isosceles triangle using the distance formula. 2. Find the point that divides the segment joining (1, 3) and (5, 7) in the ratio 1:3 internally. 3. Verify that (3, 5) is the midpoint of the segment joining (1, 3) and (5, 7).
Long Answer — 5 marks
1. The vertices of a triangle are A(2, 3), B(4, −1), C(−2, 5). Find the length of each side and determine whether it is scalene, isosceles, or equilateral. 2. Find the coordinates of the point P on the line segment joining A(3, 1) and B(−3, 7) such that AP = (2/3) AB. Also find the midpoint of AB.
Common Mistakes
- Squaring negative differences incorrectly — (x₂−x₁)² is always positive.
- Forgetting to take the square root in the distance formula.
- Mixing up m:n order in the section formula.
- Using external division formula when the question says internal division.
Revision Checklist
- [ ] Can plot points in all four quadrants
- [ ] Memorised distance formula and can apply it
- [ ] Can find midpoints using the formula
- [ ] Can apply section formula for internal division
- [ ] Can find the centroid of a triangle
- [ ] Can test collinearity of three points using distance formula