COORDINATE GEOMETRY · HL

The Plane & Midpoint

Plotting points on the Cartesian plane, and finding the midpoint.

Section 1 of 4

The Cartesian plane

Grid system $=$ Cartesian plane.

The two axes

1.$y$-axis: $x = 0$

2.$x$-axis: $y = 0$

$(0,0) =$ origin.

Section 2 of 4

Plotting points

Co-ordinates are called $(x,\ y)$.

Signs

1.$x =$ left $-$ or right $+$

2.$y =$ up $+$ or down $-$

In the door and up the stairs.

Plot $A(2,1)$ $B(-3,-2)$ $C(3,-2)$

$(2,1)$ are the co-ordinates of the point $A$.

Start at $(0,0)$. First use $2$ to go right $2$ steps. Then use $1$ to go up $1$.

Section 3 of 4

The midpoint

Plot on a Cartesian plane $P(1,4)$ and $Q(-3,2)$. Find the midpoint of $PQ$.

$\dfrac{-3+1}{2} = -1$

$\dfrac{2+4}{2} = 3$

Midpoint $= (-1,\ 3)$

Plot $A(-3,-1)$ and $B(1,3)$ on a Cartesian plane. Find midpoint of $AB$.

Midpoint $= (-1,\ 1)$

Midpoint formula

1.$\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)$

Section 4 of 4

Midpoint — worked examples

(i) Find midpoint of $(3,1)$ and $(5,9)$

$(3,1) \to x_1,y_1 \qquad (5,9) \to x_2,y_2$

$\left(\dfrac{3+5}{2},\ \dfrac{1+9}{2}\right)$

$(4,\ 5)$

(ii) Midpoint of $(7,5)$ and $(11,8)$

$\left(\dfrac{7+11}{2},\ \dfrac{5+8}{2}\right)$

$(9,\ 6.5)$

(iii) $(3,-5)$ and $(-2,-9)$

$\left(\dfrac{3-2}{2},\ \dfrac{-5-9}{2}\right)$

$(0.5,\ -7)$

(iv) Find midpoint of $(3,6)$ and $(5,2)$

$\left(\dfrac{3+5}{2},\ \dfrac{6+2}{2}\right)$

$(4,\ 4)$

SUM

The lot in one box

Plane & midpoint toolkit

1.$y$-axis: $x=0$. $x$-axis: $y=0$. $(0,0)=$ origin.

2.$x =$ left $-$ / right $+$. $y =$ up $+$ / down $-$.

3.In the door and up the stairs.

4.Midpoint $= \left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)$

End of lesson

The Plane & Midpoint — HL · Mathslive.ie