Coordinates, Distance, and Midpoint

Vocabulary

coordinate plane

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

distance formula

formula used to calculate the distance between the points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in a coordinate plane:
d=(x2x1)2+(y2y1)2d=\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 (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in a coordinate plane:
(x,y)=(x1+x22,y1+y22)\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, (x,y)\left(x, y\right), that can be used to locate a point in the coordinate plane

origin

point with coordinates (0,0)(0, 0), where the xx-axis and yy-axis intersect

Pythagorean theorem

theorem stating that if aa and bb are the legs of a right triangle and cc is the hypotenuse, then:
a2+b2=c2a^2+b^2=c^2

quadrant

one of four regions of the coordinate plane formed by the xx- and yy-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