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← Orienting Yourself: The Use of Coordinates

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Lesson 13 of 14

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Ten ideas, ten checks. The table comes first.

Ten things two numbers can tell you about a point
IdeaHow it goes
The pairFirst across, second up, from the origin. Swap them and the point changes, unless the two numbers match.
The quadrantTwo signs pick the region. A zero puts the point on an axis instead.
A shared coordinateOne subtraction. Never negative.
The distanceSquare both gaps, add, then undo the square.
The midpointAverage each coordinate on its own.
Three in a lineThe two hops cross-multiply to equal products.
A square cornerTwo squared sides add to the third, exactly.
The nameMeasure the four sides, the two diagonals, and their midpoints.
A mirrorIn the y-axis, (x, y) goes to (−x, y). In the x-axis, (x, y) goes to (x, −y). Lengths stay the same.
Against a circleCompare x² + y² with r². Less is inside, equal is on, more is outside.

A map has the well at the origin. The pump is at (3, 4). The school is at (4, 3). What do those two pairs tell you?

  • They are the same place, written two ways
  • They are equally far from the well, so it does not matter
  • They are different places, since order decides the point

Which quadrant holds the point (6, 0)?

  • The first, since 6 is positive
  • None, since it sits on the horizontal axis
  • The fourth, since the second number is zero

How far apart are (−4, 6) and (5, 6)?

  • 9, since 5 − (−4) = 9
  • 1, since 5 take away 4 is 1
  • 20, since 4 times 5 is 20

What is the distance between (−3, −4) and (4, 20)?

  • 31, since 7 across plus 24 up is 31
  • 25, since 49 + 576 = 625
  • 168, since 7 times 24 is 168

What is the midpoint of (−4, 3) and (10, 9)?

  • (7, 3), since the two gaps are halved
  • (6, 12), since you add the two pairs
  • (3, 6), since each coordinate is averaged

Are (0, 2), (3, 7) and (9, 17) in one line?

  • Yes, the two hops match
  • No, the second hop is longer than the first
  • No, the hops point in different directions

A triangle has sides 6, 5 and 5. Is any corner exactly 90°?

  • No, no two squares add to the third
  • Yes, all three lengths are whole numbers
  • Yes, one side lies flat on an axis

Corners give diagonal squares 34 and 18, with the same midpoint. What name fits best?

  • A rectangle, since opposite sides are equal
  • Nothing, since the diagonals differ in length
  • A parallelogram, since the diagonals bisect each other

The dot (−6, 2) is reflected in the y-axis. Where is its image?

  • (−6, −2), since the up number changes sign
  • (6, 2), since the across number flips
  • (6, −2), since both numbers flip

A circle has centre O and radius 6. Where is the dot (3, 5)?

  • On the circle, since 3 + 5 = 8 is near 6
  • Inside, since 9 + 25 = 34 is less than 36
  • Outside, since 3 + 5 is more than the radius 6

That is Chapter 1 of your NCERT book.

Next: Introduction to Linear Polynomials. One rule, one line, and two numbers that place it.

Reveal every answer to finish this lesson.