Coordinate Geometry · Lesson 4 of 4
Chapter Summary and Practice
“Writing a summary is basically telling a story on a strict character limit while trying not to leave out the parts where people actually care.”
• Recall the most important ideas from the chapter. • Revise the distance and section formulas. • Recognise when to use midpoint, collinearity and ratio ideas. • Practise applying coordinate geometry to different question types.
This lesson brings together the key ideas from the chapter. Use it as a quick revision sheet and then test yourself with the solved examples, practice questions and mixed check.
Coordinate Plane Basics
A point on the plane is written as an ordered pair (x, y). The x-coordinate is the abscissa and the y-coordinate is the ordinate. Points on the x-axis are of the form (x, 0), and points on the y-axis are of the form (0, y).
Distance Formula
The distance formula comes from the Pythagoras theorem. It helps us find the distance between any two points on the plane.
Section Formula
The section formula gives the coordinates of a point that divides the segment joining two given points in a given ratio internally.
Midpoint Formula
The midpoint is a special case of section formula when the ratio is 1 : 1.
Choosing the Right Formula
| What the Question Asks | Use This |
|---|---|
| Distance between two points | Distance formula |
| Distance from the origin | Special case of distance formula |
| Point dividing a segment in a ratio | Section formula |
| Midpoint of a segment | Midpoint formula |
| Whether three points are collinear | Compare distances |
| What ratio a point divides a segment in | Section formula and solve for the ratio |
Common Mistakes
• Do not change the order of coordinates. • In the distance formula, subtract x from x and y from y. • Do not forget the square root in distance questions. • In section formula, use the correct ratio order. • For midpoint, divide the sum of coordinates by 2.
Guided Practice
Problem
Find the distance between A(-2, 5) and B(4, -3).
- 1.AB = √[(4 - (-2))² + (-3 - 5)²]
- 2.= √[6² + (-8)²]
- 3.= √[36 + 64]
- 4.= √100
- 5.= 10 units.
Problem
Check whether the points P(1, 1), Q(3, 4) and R(5, 7) are collinear.
- 1.PQ = √[(3 - 1)² + (4 - 1)²] = √13.
- 2.QR = √[(5 - 3)² + (7 - 4)²] = √13.
- 3.PR = √[(5 - 1)² + (7 - 1)²] = √52 = 2√13.
- 4.Since PQ + QR = PR, the three points are collinear.
Problem
Find the point dividing the segment joining A(3, -2) and B(11, 6) in the ratio 1 : 3 internally.
- 1.x = (1×11 + 3×3) / (1 + 3) = 20/4 = 5.
- 2.y = (1×6 + 3×(-2)) / (1 + 3) = 0/4 = 0.
- 3.Therefore, the required point is (5, 0).
Problem
The point M(2, 1) divides the segment joining A(-2, -3) and B(6, 5) internally. Find AM : MB.
- 1.Let the ratio be k : 1.
- 2.Using x-coordinate: 2 = (k×6 + 1×(-2)) / (k + 1).
- 3.So, 2k + 2 = 6k - 2.
- 4.Hence, 4 = 4k, so k = 1.
- 5.Therefore, AM : MB = 1 : 1.
Problem
Find the midpoint of the segment joining C(7, -5) and D(-1, 9).
- 1.Midpoint = ((7 + (-1))/2, (-5 + 9)/2)
- 2.= (6/2, 4/2)
- 3.= (3, 2).
Quiz
Which formula do you use to find the point dividing a segment in a given ratio?
What is the midpoint of the segment joining (2, 4) and (8, 10)?
If a point lies on the x-axis, what is true about its coordinates?
Which idea helps derive the distance formula?
What ratio does the midpoint divide a segment in?
Practice Problems
- Find the distance between the pairs of points: (i) (3, 4) and (7, 1) (ii) (-6, 5) and (-1, -2) (iii) (a, b) and (-a, -b).
- Find the distance between the points (0, 0) and (24, 10).
- Determine whether the points (2, 1), (5, 5) and (8, 9) are collinear.
- Check whether (4, -1), (7, 5) and (10, -1) are the vertices of an isosceles triangle.
- Use the distance formula to decide whether the points A(1, 6), B(5, 2), C(1, -2) and D(-3, 2) form a square.
- Name the type of quadrilateral formed, if any, by the points (-2, -1), (0, 1), (-2, 3) and (-4, 1). Give reasons.
- Find the point on the x-axis which is equidistant from (3, -4) and (-1, 8).
- Find the values of y for which the distance between the points P(1, -2) and Q(9, y) is 10 units.
- If Q(0, 2) is equidistant from P(4, -2) and R(x, 7), find the values of x. Also find QR and PR.
- Find a relation between x and y such that the point (x, y) is equidistant from the points (4, 7) and (-2, 5).
- Find the coordinates of the point which divides the join of (-2, 8) and (6, -4) in the ratio 3 : 2.
- Find the coordinates of the points of trisection of the line segment joining (5, -2) and (-4, 7).
- Find the ratio in which the line segment joining the points (-4, 12) and (8, -6) is divided by the point (0, 6).
- Find the ratio in which the line segment joining A(2, -6) and B(-4, 3) is divided by the y-axis. Also find the coordinates of the point of division.
- If (2, 1), (6, y), (x, 7) and (4, 5) are the vertices of a parallelogram taken in order, find x and y.
Ask yourself: • Can I plot and read points correctly? • Can I use the distance formula confidently? • Can I check collinearity using distances? • Can I use the section formula and midpoint formula without mixing the ratios? • Can I recognise which formula a question needs?
Key Takeaways
• A point is written as (x, y), where x is the abscissa and y is the ordinate; always keep this order unchanged. • For P(x₁, y₁) and Q(x₂, y₂), the distance is PQ = √[(x₂ − x₁)² + (y₂ − y₁)²]. • The distance of P(x, y) from the origin is OP = √(x² + y²). • Use the section formula to find a point dividing a segment in a given internal ratio, and take care to match the ratio with the correct endpoints. • The midpoint formula ((x₁ + x₂)/2, (y₁ + y₂)/2) is the special case of the section formula for the ratio 1:1. • Three points are collinear when the sum of the two smaller distances between them equals the largest distance.
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Section Formula
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