Points

SAT Questions that focus on Coordinates of Points and Distance between Points require knowledge of the following topics.

Points in the Cartesian Coordinate System

The coordinates of a point in the Cartesian Coordinate System is written as two numbers in parentheses, separated by a comma, as in (2, 4), where the first number is called the abscissa (the position in the x-axis) and the second number is called the ordinate (the position in the y-axis).

For example, the following figure displays the points (-1, 3), (1, 1) e (2, 4):

Distance between two Points

Let's compute the distance between two of the points in the previous figure.

Consider "A" the point (1, 1), and "B" the point (2, 4):

Points A, B and V form a right triangle, with hypotenuse AB. According to the Pythagorean theorem:

$(AB)=\sqrt{(AV)^2+(BV)^2}$

AV is the difference between the abscissa of point B and the abscissa of point A.
BV is the difference between the ordinate of point B and the ordinate of point A.

Therefore,

$(AB)=\sqrt{(x_B-x_A)^2+(y_B-y_A)^2}$

$(AB)=\sqrt{(\Delta{x})^2+(\Delta{y})^2}$

Plugging the coordinates of points A and B,

$(AB)=\sqrt{(2-1)^2+(4-1)^2}=\sqrt{10}$

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