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How to Use the Distance and Midpoint Formulas on a Coordinate Plane

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:

distance and midpoint formula: Distance formula

The midpoint formula gives you the midpoint, expressed as an ordered pair, between the two endpoints:

Midpoint formula

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):

distance and midpoint formula: Diagram using 1-3 and 3-7 points

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:

distance and midpoint formula: Diagram showing perpendicular lines forming a right triangle

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.

distance and midpoint formula: Pythagorean theorem formula using colors

We’ll take the square root of each side:

distance and midpoint formula: Pythagorean theorem formula showing 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 x1 and y1 and (3, 7) as x2 and y2.

“Green” is the change in the x-values, so we’ll subtract the x-values of the two coordinate pairs,x1=1 and x2=3.

Likewise, the value for “blue,” the change in “y,” is y2-y1.

distance and midpoint formula: Pythagorean theorem formula subsituting the colors to its value

The distance formula emerges when we substitute these values:

distance and midpoint formula: Distance formula emerging with values being substituted

The length of the hypotenuse here is the distance between our end points. Let’s substitute the points to figure out the answer:

distance and midpoint formula: Formula showing the length of the hypotenuse

Then, we simplify according to the order of operations, PEMDAS:

distance and midpoint formula: Distance formula showing 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. x1 and y1 is (1, 3), and x2 and y2 is (3, 7).

The midpoint of the line segment is expressed as:

distance and midpoint formula: Midpoint of the line segment formula

The red dot in the middle represents the new coordinate pair for the midpoint:

distance and midpoint formula: Diagram showing the coordinate pair for the midpoint

Let’s substitute the coordinate values from above into the equation:

distance and midpoint formula: Midpoint formulas substituting the coordinate values

distance and midpoint formula: Distance formula diagram using 1-3 and 3-7 points

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 x1 and y1 and x2 and y2 is:

Midpoint formula

The distance of that same line segment denoted by coordinate pairs x1 and y1 and x2 and y2 is:

Distance formula

With a little more practice, you can navigate these formulas with ease.

More Math Homework Help:

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