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Distance formula

# Distance formula

Learn how to find the distance between two points by using the distance formula, which is an application of the Pythagorean theorem. We can rewrite the Pythagorean theorem as d=√((x_2-x_1)²+(y_2-y_1)²) to find the distance between any two points. Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

• OK, this helps a lot, but what about when the triangle does not have a right angle and it's an isosceles triangle or any other triangle?
• when dealing with graphs, this is automatically a right triangle. trig can (with a little geometry) be applied to acute or obtuse triangles.
• What should you do when you are asked to find the distance between a point and a liner equation?
• To be a bit more detailed:
1) You solve the original line equation for y if it isn't already.
2) The perpendicular line to that will be the most direct route to your point. Just take the negative inverse (if your line has a slope of 2, the negative inverse is -1/2). Which will be the slope of your perpendicular line.
3) To find the y-intercept of the perpendicular line you align it with the point you are given (if you have P(2|3) and a slope of -1/2 you can solve y=mx+c for c: 3=-1/2*2+c => c=4 and the perpendicular line will be y=-1/2x+4)
4) Then setting both lines equal you can find out where they intersect, which gets you the second point.
5) Finally you can find out the distance with Pythagoras with the distance between the points as the hypotenuse.
That's the mechanics. If you understand why you do that you have figured out almost all about linear equations.
• What is delta? (To be a little more specific.)
• Delta is a greek letter that in this case stands for change.
Delta x is the change in x. If the first point (3, 1) and the second point is (1,1), then delta x is the change in x is 3-1 or 2. Delta y is the change in y is 1=1 or 0.
• What if the points end up in a straight line, what is the correct way of solving the remainder of the equation?
• Well, for example:
-2,1 and 5,1 will be a straight line. We could just say, 'Well the length is just the change in X, since its a straight line - thats 5 - -2 = 7.' However, we want to use the distance formula.

To use the formula we need the change in x and y, dx and dy:
The Change in x is 5 - -2 = 7, dx = 7.
The change in y is 1 - 1 = 0, dy = 0.

The formula is: sqareRoot((dx*dx)+(dy*dy)).

Plugging in values, squareRoot((7*7)+(0*0))
Simplifying: squareRoot(49)
And the squareRoot of 49 is 7, or only the change in X.

You can see that either dx or dy is 0, that is the line is perfectly vertical, the formula becomes overkill :)

Hope that helps!
• Can you also use rise-over-run for this?
• Rise over run is the formula that basically describes the slope. Slope can also be calculated using the y2 - y1 / x2 - x1 formula. However, the distance formula is different. It's used to describe the length of a line segment or the distance between 2 points. Since the two formulas are used for 2 completely different things, you wouldn't be able to replace one with the other.

Good question, though. Hope this made sense!
• Can someone please tell me what hypotenuse means?
• In geometry, a hypotenuse is the longest side of a right-angled triangle, which is the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 metres (when squared, 9 m²) and the other has a length of 4 m (when squared, 16 m²), then their square
• What about when one of the points is a variable?
• You can still use the formula, and simplify like you would any other algebraic expression.
• so the distance formula = the formula of pythagorean theorem, right? because pythagorean theorem = a^2 + b^2 = c^2 and distance = d^2 = (delta x)^2 + (delta y)^2 why do people call it the distance formula eve in my school its basically the pythagorean theorem right
• Yes, it is basically a slight variation of the pythagorean theorem. It provides for finding delta X and delta Y.