Midpoint and distance between two pointsIB MYP Maths Extended: Revision notes

A Cartesian plane has a horizontal $x$-axis and a vertical $y$-axis that cross at the origin $(0, 0)$. Every point is written as an ordered pair $(x, y)$: the first number says how far across, the second how far up (or down if negative). The axes split the plane into four quadrants. A line segment is the part of a line between two points. Shapes such as triangles and quadrilaterals can be drawn by plotting their corners and joining them.

The midpoint is halfway along a segment. Take the average of the $x$-coordinates and the average of the $y$-coordinates: $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$\nExample: for $A(-2,3)$ and $B(6,7)$, $M=\left(\frac{-2+6}{2},\frac{3+7}{2}\right)=(2,5)$.

The gradient measures steepness: gradient $=\dfrac{\text{change in }y}{\text{change in }x}=\dfrac{y_2-y_1}{x_2-x_1}$. A line going up to the right has a positive gradient, down to the right a negative gradient, a horizontal line has gradient $0$ and a vertical line has no gradient (undefined). Example: from $(1,1)$ to $(4,5)$, gradient $=\frac{5-1}{4-1}=\frac43$. Three points are on one straight line if the gradients between them are equal.

The horizontal and vertical differences between two points form the two shorter sides of a right-angled triangle, and the segment is the hypotenuse. So $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.$\nExample: $A(-2,3)$ to $B(6,7)$: differences $8$ and $4$, so $d=\sqrt{64+16}=\sqrt{80}=4\sqrt5\approx8.94$. Leave answers as exact surds unless asked to round. Distance, speed and time link as time $=\dfrac{\text{distance}}{\text{speed}}$.

Use lengths and gradients to identify shapes. Isosceles: two equal sides. A right angle exists if $a^2+b^2=c^2$ for the three sides (the converse of Pythagoras). A quadrilateral with four equal sides and a right angle is a square. For area, use a horizontal or vertical side as the base: for $P(1,1)$, $Q(7,1)$, $R(4,5)$, $PQ=6$ and the height is $5-1=4$, so area $=\frac12\times6\times4=12$. The midpoint of a segment can be used to find the centre of a shape or a halfway point on a route.