With the distance and midpoint formulas, you can find the distance and midpoint between any two points on a coordinate plane.
The distance formula gives you the distance d, expressed as a single value, between the two endpoints:
The midpoint formula gives you the midpoint, expressed as an ordered pair, between the two endpoints:
How to Solve the Distance Formula
The distance formula works for any two given points. Let's take a look at this example with coordinate points (1, 3) and (3, 7):
If we imagine the two end points as two vertices on a triangle, we can see how the distance formula works. Remember that the hypotenuse of a right triangle, when squared, equals the sum of the square of the two legs. This is where the distance formula comes from.
Here’s where the triangle is in our diagram:
In this example, the green and blue dotted sides of the triangle are perpendicular lines that form a right triangle.
This blue side of the triangle is 4 units long, and the green side of the triangle is 2 units long, as we can see on the graph. But how long is the red side? It’s the distance between our two original points.
Let’s apply the Pythagorean theorem, c² = a² + b². We’ll substitute the red line for c, the hypotenuse, and the green and blue lines for sides a and b.
We’ll take the square root of each side:
Finally, we substitute the colors. “Red” becomes D, or the distance.
Now, look at our coordinate pairs. We’ll refer to (1, 3) as and (3, 7) as .
“Green” is the change in the x-values, so we’ll subtract the x-values of the two coordinate pairs,.
Likewise, the value for “blue,” the change in “y,” is .
The distance formula emerges when we substitute these values:
The length of the hypotenuse here is the distance between our end points. Let’s substitute the points to figure out the answer:
Then, we simplify according to the order of operations, PEMDAS:
We can leave it as a square root or simplify it to numeric value, 4.47.
How to Solve the Midpoint Formula
You’ll use the midpoint formula to find the middle point between any two points. To do this, the formula looks at the coordinates of the endpoints and then finds the average value between the x-coordinates and the y-coordinates.
In the graph below, we have a line segment between the two coordinate points. is (1, 3), and is (3, 7).
The midpoint of the line segment is expressed as:
The red dot in the middle represents the new coordinate pair for the midpoint:
Let’s substitute the coordinate values from above into the equation:
Applying the Distance and Midpoint Formulas
The distance and midpoint formulas can help us find the distance and midpoint between two endpoints on a coordinate plane. The midpoint of a line segment formula between two coordinate pairs and is:
The distance of that same line segment denoted by coordinate pairs and is:
With a little more practice, you can navigate these formulas with ease.