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Distance formula review

Review the distance formula and how to apply it to solve problems.

What is the distance formula?

The formula gives the distance between two points left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis and left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis on the coordinate plane:
square root of, left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, right parenthesis, squared, plus, left parenthesis, start color #e07d10, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis, squared, end square root
It is derived from the Pythagorean theorem.
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points. A third unlabeled point is at x two, y one with a line connecting from it to the point at x two, y two and another line connecting from it to the point at x one, y one forming a right triangle. The hypotenuse of the right triangle is unknown and the side made from the point at x one, y one and x two, y one is labeled x two minus x one. The third side is labeled y two minus y one.
Want to learn more about the distance formula? Check out this video.

What problems can I solve with the distance formula?

Given two points on the plane, you can find their distance. For example, let's find the distance between left parenthesis, start color #1fab54, 1, end color #1fab54, comma, start color #e07d10, 2, end color #e07d10, right parenthesis and left parenthesis, start color #1fab54, 9, end color #1fab54, comma, start color #e07d10, 8, end color #e07d10, right parenthesis:
=(x2x1)2+(y2y1)2=(91)2+(82)2Plug in coordinates=82+62=100=10\begin{aligned} &\phantom{=}\sqrt{(\greenD{x_2 - x_1})^2 + (\goldD{y_2 - y_1})^2} \\\\ &=\sqrt{(\greenD{9 -1})^2 + (\goldD{8 - 2})^2}\quad\small\gray{\text{Plug in coordinates}} \\\\ &=\sqrt{8^2+6^2} \\\\ &=\sqrt{100} \\\\ &=10 \end{aligned}
Notice: we were careful to put the x-coordinates together and the y-coordinates together and not mix them up.

Check your understanding

Problem 1
What is the distance between left parenthesis, 4, comma, 2, right parenthesis and left parenthesis, 8, comma, 5, right parenthesis?
Choose 1 answer:
Choose 1 answer:

Want to try more problems like this? Check out this exercise.

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