To recall, real numbers consist of integers, fractions, rational and irrational numbers which we can represent on the real number line. When we move a positive direction on the number line it is almost always moving up or two the right, when we move in a negative direction it is almost always to the left.
On the number line, if we move from x = a to x = b then distance moved is b - a. This change of x is symbolized by Δx.
This symbol, (delta) Δ is what we use to symbolize a change in something. So,
For moving from x = a to x = b, the change in x, a to b is represented like,
Δx = b - a
If x = b is a bigger value than x = a, then Δx = b - a will be positive, meaning that the move from a to b is in a positive direction.
If x = b is a smaller value than x = a, then Δx = b - a will be negative, meaning that the move from a to b is in a negative direction.
If x = b is the same as x = a, then Δx = b - a will be zero, meaning that there was no movement of the points.
We also use Δ when representing the distance between two points a and b, for this we take the absolute value of Δ, or |Δ|, meaning strictly the positive value of the number. ( |-3| = 3 )
dist(a, b) = b - a if b ≥ a or a - b if b > a
Or, using our absolute value (the positive value),
The Midpoint is the point halfway between two points, we calculate this by,
Question: Find the distance and the midpoint of the points x = -1 and x = 7,
Recall our definition, the distance between two points, a and b, on a number line is,
We want to know how far away x = -1 is from x = 7,
[if we just calculated Δ x then that would give us the direction (if it was positive or negative) as well as the distance, which we don't want.]
For this we want the absolute value of Δ x, that is, the positive value of Δ x.
dist(-1, 7) = |delta x| = |7 - (-1)| = |8| = 8
So the distance is 8
For the Midpoint,
M = 3
So the midpoint between x = -1 and x = 7 is x = 3
A Cartesian plane is defined by two perpendicular number lines: the x-axis, which is horizontal, and the y-axis, which is vertical. Using these axes, we can describe any point in the plane using an ordered pair of numbers, usually x and y, said like (x, y). A Cartesian plane is thought of as two dimensional.
If we have to points, A at (x1, y1) and B at (x2, y2) on a Cartesian plane and we want to calculate the distance between them, we approach it the same way we would when calculating two points on a number line, except we have to account for the fact we have two dimensions rather than one.
The horizontal direction of the change in x-values is:
Δ x = x2 - x1
The vertical direction of the change in y-values is
Δ y = y2 - y1
For our distance between A and B recall the Pythagorean theorem of,
a2 + b2 = c 2,
In this case, the c value from the theorem is equivalent to dist(A, B), and the Δ x and Δ y are equivalent to the a and b value from the theorem,
(dist(A, B))2 = (Δ x)2 + (Δ y)2
The Midpoint, M of A and B is:
Question: Calculate the distance and midpoint between points T (2, 3) and R (8, 10) to the nearest tenth of a decimal place
So, our equation for distance between two points on two dimensional Cartesian plane is,
The midpoint is,
Question: Find an equation for the line with all the points of P = (x, y) that are an equal distance from S = (1, 2) and T = (4, -2).
For this question, we are looking for all the points that are a equal distance from the points S and T, these points will all be on a straight line,
Note, that for every point on the line with all the equal distant points, each point on that line is the same distance from S as it is from T (d1, d2, d3 etc…).
Any point that is not on that line will be closer to either the point T or S.
Now, lets do the math,
We know that the general formula for distance between two points A(x1, y1) and B(x2, y2) is,
And we know that all the points P(x, y) are the same distance from S as they are from T,
The distance from P to T is equal to the distance from P to S, so,
dist(P, T) = dist(P, S)
Now, we sub in the values of P(x, y) - which will give us a general equation for the straight line between the two points T and S. We also sub in our T(4, -2) and S(1, 2) into,
dist(P, T) = dist(P, S)
Now equating those two equations,
Now, squaring both sides of the equation we get,
(x - 4)2 + (y + 2)2 = (x - 1)2 + (y - 2)2
x2 -8x + 16 + y2 + 4y + 4 = x2 - 2x + 1 + y2 - 4y + 4
Subtract x2 and y2 from both sides of the equation,
-8x + 16 + 4y + 4 = - 2x + 1 - 4y + 4
-8x + 20 + 4y = -2x + 5 - 4y
Add 2y to both sides,
-8x + 20 + 8y = -2x + 5
Add 8x and subtract 20,
8y = 6x - 15
Divide both sides by 8,
There we have our equation for the line that runs between the points S and T, every point on that line will be an equal distance from S and T.