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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:
\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.

Want to join the conversation?

• The soccer field question confuses me - I don't understand how =√52 becomes 2√​13!
Any help?
(32 votes)
• To simplify √52, we look for the largest perfect square factor of 52. This factor is 4.
Since the square root of a product of positive numbers is the product of their square roots,
√52 = √(4*13) =√4 * √13 = 2√13.
(85 votes)
• what is the Pythagorean therom
(2 votes)
• The pythagorean therom states that a^2 + b^2 = c^2. Or, the sum of the two short sides of the right triangle squared = the hypotunuse (the long side of the right triangle) squared

NOTE: this works and ONLY WORKS FOR A RIGHT TRIANGLE. It does NOT and CAN'T apply to any other type of triangle
(12 votes)
• how do u solve it if the 2 x's are negetavie
(5 votes)
• You can still subtract a negative from a negative. -3-(-6) is the same as -3+6, which is three. If you instead do -6-(-3), it's the same as -6+3, which is -3. Either works as you will be squaring it, which is a positive number anyways. Hope this helps!
(1 vote)
• Dont understand the graph
(4 votes)
• The graph is showing the x and y coordinates of the three points so that you can find the differences in x and y. Once you have found the difference in x and y, you can then solve using the Pythagorean Theorem.
(2 votes)
• Hello, I Don't understand the distance between(2,2) and (4,7) I'm very confused and i'm good at math but wow.
Any help?
(1 vote)
• The two points are (𝑥₁, 𝑦₁) = (2, 2) and (𝑥₂, 𝑦₂) = (4, 7)

The distance formula tells us
𝑑 = √((𝑥₁ − 𝑥₂)² + (𝑦₁ − 𝑦₂)²)
= √((2 − 4)² + (2 − 7)²)
= √((−2)² + (−5)²)
= √(4 + 25)
= √29
(8 votes)
• how do you use the Pythagorean Theorem
(2 votes)
• The Pythagorean Theorem is used to find the side length of RIGHT triangles. THIS CAN NOT BE USED FOR ANY OTHER TYPE OF TRIANGLE! You take the length of side a(the side that creates a right angle) and square it. Then add it to side b squared(the other side creating a right angle) and this will give you the value of c sqaured. Now square root this value to get the length of side c, or the hypotenuse, which is perpendicular to the right angle. Hope this helps!
(4 votes)
• It all confuses me. But mostly the soccer feild.
(3 votes)
• Am I allowed to use trigonometry to solve these?
(1 vote)
• If you use trig, it will be a more indirect way to get the answer, but you can if you want. You would have to find the inverse tan to find the angle, then apply either sin or cos to find the third side.
(3 votes)
• im just confused how 52 turned into 2times the square route of 13. it doesnt explain why, it just expects you to understand it.
(2 votes)
• I believe you meant the square root of 52, not 52 itself.

Because 52=4*13 and the square root of a product of non-negative numbers is always the product of the square roots, we have

sqrt(52) = sqrt(4*13) = sqrt(4)*sqrt(13) = 2sqrt(13).
(1 vote)
• i understood all of them except the last one i got 52 then square root it and it was 7.2111 so i just guessed between those two i picked
(2 votes)