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CBSE Class 10 · Mathematics

Coordinate Geometry

Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.

This lesson follows the official chapter “Coordinate Geometry” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 10
  • Medium level
  • 8 concepts

1Distance Formula

Distance between (x1, y1) and (x2, y2) is √[(x2−x1)²+(y2−y1)²] — the hypotenuse of |Δx| and |Δy|. From (2, 3) to (6, 6) is √(16+9)=5. A walk of 4+3 is 7, not this distance.

3+4 = 7 is the Manhattan steal.

Figure. Distance is the hypotenuse of the right triangle whose legs are the run and the rise. From A(1, 2) to B(4, 6) those legs are 3 and 4, so d = √(3² + 4²) = 5.

How it works

  1. Subtract the xs and the ysThe legs.
  2. Square, add, square-rootThe hypotenuse.
  3. Keep the unit if the grid has oneThe distance.

5 on the plane

Distance from (2, 3) to (6, 6).

  • Δx4
  • Δy3
  • √(16+9)5

Pro tip. The plane-distance is the right-triangle hypotenuse.

2Section Formula

The section formula splits a segment in the ratio m:n. The point dividing A(x1,y1) and B(x2,y2) in m:n (as taught, internally) is ((n x1 + m x2)/(m+n), (n y1 + m y2)/(m+n)). Midpoint is the 1:1 case: averages. From (2, 3) to (6, 6), midpoint is (4, 4.5).

Using m:n as a subtract instead of a weighted average is a miss.

Figure. P splits AB internally in 2 : 1, so it sits two-thirds of the way from A to B: P = ((2·8 + 1·2)/3, (2·4 + 1·1)/3) = (6, 3). The 2-part span is twice the 1-part span.

How it works

  1. Name m:n and the two endsThe given.
  2. Write the weighted average as taughtThe section.
  3. For 1:1, average the xs and the ysMidpoint.

3A definition is a test you can run

Distance / section is a test: two points plus a formula. If you cannot name the pairs, you have a doodle.

A city-map without axes is a different locate.

Figure. Collinear is a test, not a look: AB + BC equals the long side AC. Here AB = √5, BC = 2√5, AC = 3√5, so the two short hops add to the long hop and the three points sit on one line.

How it works

  1. Name the two pairs (and m:n if asked)The object.
  2. Run distance or the weighted averageThe test.
  3. Then the word has contentThe definition ran.

4Name the given before the unknown

The given are the two points (and m:n). The unknown is the distance or the section-point. Copy the order (x, y) before you subtract.

Swapping 6 and 3 because the picture “looks taller” is a silent given.

Figure. Write the two given pairs before you reach for √(Δx² + Δy²). Here the given are A(3, 1) and B(7, 5). The dashed join is the unknown length, not a number you invent first.

How it works

  1. Copy the pairs and the ratioThe given.
  2. Name distance or the pointThe unknown.
  3. Then run the formulaGiven first.

5One worked case is enough at this class

One distance or one midpoint is enough. A page of the same 5-clone does not add a new section.

Ten twins of (4, 4.5) are still one idea.

Figure. One school case is enough to lock the move: A(2, 1) to B(6, 4) has run 4 and rise 3, so d = √(16 + 9) = 5. Same triangle as 3-4-5, just translated off the origin.

How it works

  1. Compute one distance or one sectionThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

6A miss: swapping the school name for the picture

A treasure-map photo is a setting. The maths is (x, y) and a formula. An X-mark is not a midpoint until the average is run.

A pirate still is not the section formula.

Figure. The pair is an address, not a chapter title. (4, 2) sits 4 along x and 2 up. Swapping the ink to (2, 4) lands on a different point. Draw the axes before you name the formula.

How it works

  1. Keep the map as a storyA setting.
  2. Write the pairs and m:nThe maths.
  3. Then distance or the pointDo not swap.

7Check by the opposite action or the opposite test

Check a distance by swapping the two points — it must match — or by a 3-4-5 read. Check a midpoint by seeing it sits halfway on each axis.

Counting Manhattan blocks is not the opposite test for this formula.

Figure. The opposite test of a midpoint claim is AM = MB. Midpoint of A(2, 3) and B(8, 7) is M(5, 5). Each hop is (3, 2), so the two distances match.

How it works

  1. State 5 or (4, 4.5)The claim.
  2. Swap ends or average againThe check.
  3. Mend if they disagreeHonest.

8Keep the claim at this chapter, not the next

This chapter’s claim stops at distance and section. It does not steal sine-opposite-hypotenuse, and it does not start a 3-d distance.

A page that already “defines tan” here misses the trigonometry chapter.

Figure. Class 10 locks distance and the section/midpoint split. The equation of a line is a later official chapter — a dashed hop, not a tool you import here.

How it works

  1. Keep √(Δx²+Δy²) and the weighted pointThis chapter.
  2. Leave ratios of sides for trigonometryThe next map.
  3. Do not treat the map as already answeredClaim-size.
Distance from (2, 3) to (6, 6) is
  1. 5
  2. 7
  3. 4

√(16+9)=5, not 4+3.

Notes

  • The official chapter title is “Coordinate Geometry”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • distance = √[(x2−x1)²+(y2−y1)²]
  • midpoint = ((x1+x2)/2, (y1+y2)/2)

Recap

Hold these pegs from the official chapter “Coordinate Geometry”. The wording is ExamMaster’s teaching, not a textbook recap.

Distance Formula
Distance between (x1, y1) and (x2, y2) is √[(x2−x1)²+(y2−y1)²] — the hypotenuse of |Δx| and |Δy|.
Section Formula
The section formula splits a segment in the ratio m:n.
A definition is a test you can run
Distance / section is a test: two points plus a formula.
Name the given before the unknown
The given are the two points (and m:n). The unknown is the distance or the section-point. Copy the order (x, y) before you subtract.
One worked case is enough at this class
One distance or one midpoint is enough. A page of the same 5-clone does not add a new section.
A miss: swapping the school name for the picture
A treasure-map photo is a setting. The maths is (x, y) and a formula. An X-mark is not a midpoint until the average is run.

Practise Coordinate Geometry

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