Coordinates

Coordinates

A coordinate system uses two perpendicular number lines (axes) to locate every point in a plane. The horizontal axis is the \( x \)-axis; the vertical is the \( y \)-axis. Every point has a unique pair of coordinates \( (x, y) \) — the signed distances from the origin along each axis. The axes divide the plane into four quadrants.

x y 0 −4 −2 2 4 4 2 −2 −4 I II III IV A(3, 2) B(−3, 2) C(−3,−2) D(3,−2)

Midpoint of a Line Segment

The midpoint of the segment joining \( (x_1, y_1) \) to \( (x_2, y_2) \) is the average of each coordinate:

\[ M = \left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right) \]

Example: midpoint of \( (1, 3) \) and \( (5, 7) \) is \( \left(\frac{6}{2}, \frac{10}{2}\right) = (3, 5) \).

Distance Between Two Points

The distance uses Pythagoras's theorem on the horizontal and vertical separations:

\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]

Example: distance from \( (1, 2) \) to \( (4, 6) \) is \( \sqrt{3^2+4^2} = \sqrt{9+16} = \sqrt{25} = 5 \).

Worked Examples

State the quadrant of each point: \( (5, -2) \), \( (-4, -1) \), \( (-3, 6) \), \( (0, 3) \).

\( (5, -2) \): \(x>0, y<0\) — Quadrant IV.

\( (-4, -1) \): \(x<0, y<0\) — Quadrant III.

\( (-3, 6) \): \(x<0, y>0\) — Quadrant II.

\( (0, 3) \): on the \(y\)-axis — not in any quadrant.

Find the midpoint of \( A(-4, 3) \) and \( B(6, -1) \). Then find \( B \) if the midpoint is \( (2, 4) \) and \( A = (-1, 7) \).
\[ M = \left(\frac{-4+6}{2},\ \frac{3+(-1)}{2}\right) = (1, 1) \]

For the second part: \( \frac{-1+B_x}{2} = 2 \Rightarrow B_x = 5 \); \( \frac{7+B_y}{2} = 4 \Rightarrow B_y = 1 \). So \( B = (5, 1) \).

Find the exact distance from \( P(-2, 1) \) to \( Q(3, 4) \). Then verify that triangle \( A(0,0) \), \( B(4,0) \), \( C(0,3) \) is right-angled.
\[ PQ = \sqrt{(3-(-2))^2+(4-1)^2} = \sqrt{25+9} = \sqrt{34} \]

For the triangle: \( AB = 4, AC = 3, BC = \sqrt{16+9} = 5 \). Check: \( 3^2+4^2 = 25 = 5^2 \checkmark \) — right angle at \( A \).

 Key Takeaways

  • Coordinates \( (x, y) \): \( x \) is the horizontal position (positive right), \( y \) is the vertical (positive up).
  • Quadrant I: \( (+,+) \); II: \( (-,+) \); III: \( (-,-) \); IV: \( (+,-) \). Points on axes are not in any quadrant.
  • Midpoint: average the \( x \)-coordinates and average the \( y \)-coordinates separately.
  • Distance: \( d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \) — Pythagoras applied to the horizontal and vertical separations.
  • To find an unknown endpoint given the midpoint: double each midpoint coordinate then subtract the known endpoint.