7. Coordinate Geometry

5 topics1 h

Chapter 7 of NCERT Mathematics for Class 10: Coordinate Geometry. Two sections — How far apart are two points and Cutting a segment in a given ratio. Five videos, 1 h in all.

Worked answers to the 20 exercise questions in this chapter: Exercise 7.1 · Exercise 7.2

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Part 1

How far apart are two points

14 minAbscissa and ordinate, and what a coordinate pair actually recordsSay five. Which point? One number names a whole line's worth of places when what you needed was one place - and that is the entire reason a coordinate pair carries two entries, each of them measured from the ruler it is not named after.14 min · 12 parts15 minDeriving the distance rule by hanging a right triangle off the two pointsTwo points, a slant between them, and no right angle anywhere in the question. So we put one there - three lines that were not in the problem, drawn for the single reason that a right triangle is a thing we already know how to finish.15 min · 14 parts15 minUsing distances alone to classify a triangle or a quadrilateralFour equal sides do not make a square. They make a rhombus - and the reason that matters is not the rhombus, it is that a classification needs enough independent measurements to force the definition shut, and counting how many is the whole skill.15 min · 14 parts

Part 2

Cutting a segment in a given ratio

15 minGetting the section rule from a pair of similar trianglesA ratio is not a fraction of the whole, and the weights in the section rule come out crossed. Both of those are one idea: the two pieces are compared with each other, and a bigger near piece is exactly what pushes the point further away.15 min · 14 parts16 minThe midpoint as the ratio 1 : 1, and recovering an unknown ratioThe midpoint rule is the section rule with one substitution made, and running it backwards needs one coordinate, not two - any coordinate in which the two endpoints actually differ.16 min · 14 parts

What you will be able to do

  • Say which axis each of the two coordinates is measured from, and use the printed names abscissa and ordinate correctly
  • Set up the auxiliary construction for two given points — two perpendiculars to the horizontal axis and one horizontal segment — and name the right angle it creates
  • Compute all three pairwise separations for a triple of points and decide from them whether the points form a triangle or lie on one line
  • Restate a positional requirement given in words as a ratio in which a point cuts a segment
  • Derive the mid-point expressions by putting equal parts into the section rule, rather than quoting them

What this chapter assumes you already know

  • Plotting a point on squared paper from an ordered pair, and reading a pair back off a plotted point (Class IX coordinate geometry, which §7.1 names as the starting point)
  • Negative numbers on a number line, and subtracting a larger value from a smaller one
  • A linear equation in two variables has a graph, and that graph is a straight line
  • Square roots of small whole numbers, and the idea that a square root of a non-negative quantity can be taken as non-negative
  • Perpendicular lines, and dropping a perpendicular from a point to a line
  • Pythagoras' theorem in the form: on a right triangle, the square on the side facing the right angle equals the sum of the squares on the other two
  • That a square of a number is non-negative, whatever the sign of the number
  • Simplifying surds far enough to write √8 as 2√2 and √50 as 5√2

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "The x-coordinate is the distance from the x-axis."
  • "(3, 5) and (5, 3) are the same point."
  • "A coordinate is a distance, so it cannot be negative."
  • "A point sitting on the x-axis has no y-coordinate."
  • "Coordinate geometry is a way of drawing graphs."
  • "You can always subtract coordinates to get a separation."
  • "You subtract the coordinates and that is the distance."
  • "You have to put the smaller coordinate first."