Coordinates, Distance, and Midpoint

Vocabulary

coordinate plane

plane used to locate points by using coordinates $\left(x, y\right)$ that give the distance from the $x$-axis and the $y$-axis

distance formula

formula used to calculate the distance between the points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane:
$d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$

midpoint

point that lies on a line segment connecting two points and divides the line segment into two equal halves

midpoint formula

formula used to calculate the coordinates of the midpoint between points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane:
$\left(x,\;y\right)=\left(\frac{x_{1}+x_{2}}2,\;\frac{y_{1}+y_{2}}2\;\right)$

ordered pair

pair of numbers written in order, $\left(x, y\right)$, that can be used to locate a point in the coordinate plane

origin

point with coordinates $(0, 0)$, where the $x$-axis and $y$-axis intersect

Pythagorean theorem

theorem stating that if $a$ and $b$ are the legs of a right triangle and $c$ is the hypotenuse, then:
$a^2+b^2=c^2$

one of four regions of the coordinate plane formed by the $x$- and $y$-axes. The quadrants are numbered I, II, III, and IV, counterclockwise beginning in the top right quadrant.

x-axis

horizontal number line or axis in the coordinate plane

x-coordinate

first value in an ordered pair of coordinates, which determines the horizontal distance of a point from the origin

y-axis

vertical number line or axis in the coordinate plane

y-coordinate

second value in an ordered pair of coordinates, which determines the vertical distance of a point from the origin