Distance Formula | Calculator & Step By Step Examples
The Distance Formula is a variant of the Pythagorean Theorem that you used back in geometry. Here's how we get from the one to the other: Suppose you're given the two points (–2, 1) and (1, 5), and they want you to find out how far apart they are. The distance formula is derived from the Pythagorean theorem. To find the distance between two points (x 1, y 1) and (x 2, y 2), all that you need to do is use the coordinates of these ordered pairs and apply the formula pictured below. The distance formula is Distance = (x 2 ? x 1) 2 + (y 2 ? y 1) 2.
Formula Examples. The Distance Formula is a variant of the Pythagorean Theorem that you used back in geometry. Here's how we get from the one to the other:. Suppose you're given the two points —2, 1 and 1, 5and they want you to find out how far apart they are. The points look like this:. Distance Formula. You can draw in the lines that form a right-angled triangle, using these points as two of the corners:. It's easy to find what species has the largest population on earth lengths of the horizontal and vertical sides of the right triangle: just subtract the x -values and the y -values:.
Then use the Pythagorean Theorem to find the length of the third side which is the hypotenuse of the right triangle :. This format always holds true. Given two points, you can always plot them, draw the right triangle, and then find the length of the hypotenuse.
The length of the hypotenuse is the distance between the two points. Since this format always works, it can be turned into a formula:. Distance Formula: Given the two points x 1y 1 and x 2y 2the distance d between these points is given by the formula:. Don't let the subscripts scare you. They only indicate that there is a "first" point and a "second" point; that is, that you have two points. Whichever one you call "first" or "second" is up to you. The distance will be the same, regardless.
Then the distance is sqrt 53or about 7. Page 1 Page 2. All right reserved. Web Design by. Skip to main content. The Distance Formula Formula Examples. Purplemath The Distance Formula is a variant of the Pythagorean Theorem that you used back in geometry.
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The Pythagorean Theorem and the distance formula
To find the distance between to points if it is just a line, you simply draw a dot where the line ends, then you make a number line and find the coordinates for both points. Finally, you follow the distance formula, plug the values in, and solve. I hope this helps and wasn't a bore! (1 vote). Distance Formula In Figure 1, A is (2, 2), B is (5, 2), and C is (5, 6). Figure 1 Finding the distance from A to C. To find AB or BC, only simple subtracting is necessary. Distance formula The distance formula is a formula that determines the distance between two points in a coordinate system. Distance formula for a 2D coordinate plane: Where (x 1, y 1) and (x 2, y 2) are the coordinates of the two points involved.
How it works: Type the two x coordinates and two y coordinates into the boxes below and it will automatically calculate the distance between those 2 points and show you step by step.
In a Cartesian grid, to measure a line segment that is either vertical or horizontal is simple enough. You can count the distance either up and down the y-axis or across the x-axis. But what about diagonal lines? How can you know precisely how long the line segment is if it cuts across those tiny boxes? See this example:.
You can use the Distance Formula to calculate any line segment if you know the coordinates of the two endpoints. You will be mentally constructing a right triangle, using the diagonal as if it were a hypotenuse. The Distance Formula squares the differences between the two x coordinates and two y coordinates, then adds those squares, and finally takes their square root to get the total distance along the diagonal line:. The expression x 2 - x 1 is read as the change in x and y 2 - y 1 is the change in y.
What this is really doing is calculating the distance horizontally between x values, as if a line segment was forming a side of a right triangle, and then doing that again with the y values, as if a vertical line segment was the second side of a right triangle. That leaves a calculation about the hypotenuse, your given diagonal. The Distance Formula is a special application of the Pythagorean Theorem.
All you need to do is plug the coordinates in very carefully. Let's use our line's endpoints, 1 , 3 and 7 , 6 :. You need not even have a coordinate grid in front of you to use the Distance Formula, so long as you have both sets of coordinate points.
So, try these three practice problems! We will not leave you hanging out on a diagonal. Here are the beginning steps, to help you get started:. The Distance Formula gets its precision and perfection from the concept of using the angled line segment as if it were the hypotenuse of a right triangle formed on the grid. You need not construct the other two sides to apply the Distance Formula, but you can see those two "sides" in the differences distances between x values a horizontal line and y values a vertical line.
Now that you have worked through the lesson and practice, you are able to apply the Distance Formula to the endpoints of any diagonal line segment appearing in a coordinate, or Cartesian, grid.
You are also able to relate the Distance Formula to the Pythagorean Theorem. Pythagoras was a generous and brilliant mathematician, no doubt, but he did not make the great leap to applying the Pythagorean Theorem to coordinate grids.
His Cartesian grid combines geometry and algebra. You can use formulas, including the Distance Formula, to get precise measurements of line segments on the grid. What is a Straight Line?
Get better grades with tutoring from top-rated professional tutors. Get help fast. Want to see the math tutors near you? Distance Formula Calculator. Auto Calculate. See this example: You can use the Distance Formula to calculate any line segment if you know the coordinates of the two endpoints.
Distance Formula The Distance Formula squares the differences between the two x coordinates and two y coordinates, then adds those squares, and finally takes their square root to get the total distance along the diagonal line:.
What you'll learn: After working your way through this lesson and video, you will be able to: Apply the Distance Formula to the endpoints of any diagonal line segment appearing in a coordinate, or Cartesian, grid Relate the Distance Formula to the Pythagorean Theorem. Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher. Local and online. View Tutors. Geometry Help. Analytic Geometry. Tutors online. Ask a question Get Help. View Math Tutors. Popular cities for math tutoring Math Tutors New York. Find a math tutor near you Learn faster with a math tutor.
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