Main content

## High school geometry

### Course: High school geometry > Unit 6

Lesson 1: Distance and midpoints# 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:

It is derived from the Pythagorean theorem.

*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:

Notice: we were careful to put the x-coordinates together and the y-coordinates together and not mix them up.

## Want to join the conversation?

- The soccer field question confuses me - I don't understand how =√52 becomes 2√13!

Any help?(35 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.(99 votes)

- these comments are actually so nice(11 votes)
- what is the Pythagorean therom(3 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(17 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(12 votes)

- I have a question about the last problem. Simplify one of the distances and the final formula is 2√13 + √13, but I didn't know the same number of square roots can be combined. Does anyone know where I can learn about it?(2 votes)
- anytime you can substitute an x, you may be able to simplify. Let x=√13, this gives 2x + x, and hopefully you understand by now that you can combine like terns to get 3x. Substitutin back in, you get 3√13.(8 votes)

- 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)

- Am I allowed to use trigonometry to solve these?(0 votes)
- 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.(9 votes)

- It all confuses me. But mostly the soccer feild.(4 votes)
- how do u solve it if the 2 x's are negetavie(4 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!(0 votes)

- What's the best way to remember the Distance formula?(2 votes)
- The best way to remember the distance formula is to observe that it is the result of using the Pythagorean theorem.

Imagine a right triangle with vertices at (x_1, y_1), (x_2, y_1), and (x_2, y_2). The legs have lengths |x_2 - x_1| and |y_2 - y_1|, and the hypotenuse has length equal to the distance from (x_1, x_2) to (y_1, y_2). Using the Pythagorean theorem on this right triangle yields the distance formula.

Have a blessed, wonderful day!(3 votes)