A point P is positioned relative to two perpendicular lines, called the coordinate axes. The perpendicular distance from the y-axis to point P is the x-coordinate; the perpendicular distance from the x-axis to point P is the y-coordinate. The coordinates x and y form an ordered pair (x, y).
Often, a grid is used to display points relative to the coordinate axes.
The point (4, 3) is located 4 units from the y-axis to the right and 3 units above the x-axis; the point (-2, 1) is 2 units to the left of the y-axis and 1 unit above the x-axis. The distance, d, between the two points can be found by the Pythagorean Theorem. The horizontal leg is the total distance in the x-direction: 4 - (-2) = 6; the vertical leg is the distance in the y-direction: 3 - 1 = 2. The distance is then
d = 6 + 2=
= 2
Example 1 (easy)
A square has two corners of a diagonal at (6, 8) and (2, 4). What is its area?
Solution
Compare the x1 and x2 values and the y1 and y2 values. The difference in the x-direction is 6 - 2 = 4 and in the y-direction 8 - 4 = 4. The sides are both of length 4, so that the area is A = 4 × 4 = 16.
Slope Formula
How do you measure the slant of a line? By definition, it is the ratio of the vertical change to the horizontal change (see figure below).
Forming the vertical change over the horizontal change (above) figure results in slope formula (where m is the slope).
Use this formula to calculate slopes of lines.
Example 2 (easy)
What is the slope of this line?
Solution
To solve this problem, plug the digits in the line into the slope intercept equation.
y = 4, b = 2
x = 5, a = 1
The slope is 1/2.
Slope Intercept Formula
If you have a formula, such as x - 2y = 4, how do you calculate the slope of the line?
If you want to graph a line, the formula to use is:
y = mx + b
In this equation, m is the slope of the line and b is the y-intercept.
The y-intercept is when x = 0 in an equation.
The x-intercept is when y = 0 in an equation.
The x-intercept is the point where the line crosses the x-axis. It is found by setting y = 0 and solving the resulting equation. The y-intercept is the point where the line crosses the y-axis. It is found by setting x = 0 and solving the resulting equation.
Example 3 (easy)
Graph the equation 4x - y = 5.
Solution
Try to convert the equation 4x - y = 5 into the format
y = mx + b
4x - y = 5 can be converted into
y = 4x - 5
This means that m = 4 (the slope is 4) and b = -5 (the y-intercept is -5)
In the above graph, the slope is 4 and the y-intercept is -5.
In the above graph, the dot is the y-intercept at (0, -5).
The line slopes up at a rate of 4 up for every 1 across (slope of 4). It intersects lines at (1,-1) and (2,3).
Distance Formula
The distance formula is an adaptation of the
Pythagorean Theorem which is used to find the distance between two
points on the coordinate plane. The formula states:
d = √(X2 - X1)2 + √(Y2 - Y1)2
Example 4 (medium)
What is the distance between the points (3, 6)
and (4, 7) on the coordinate plane?
Solution
In this problem, there are two ways to figure out the distance.
One is to draw a triangle using these points where the line from (3,6) to (4,7) is the hypotenuse, a line from (3,6) to (4,6) is one leg, and a line from (4,6) to (4,7) is another leg. From this, find out the distance of each leg, plug it into the Pythagorean theorem and you get that the hypotenuse is equal to √2.
The other way to approach this problem is to use the distance formula on the two points given.
d = √(4 - 3)2 + √(7 - 6)2
d = √(1)2 + √(1)2
d = √(2)
Both ways work to give the same answer, but sometimes the distance formula is a much more direct way to figure out the distance between two points.