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### Course: Geometry (FL B.E.S.T.)>Unit 1

Lesson 3: 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 $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ on the coordinate plane:
$\sqrt{\left({x}_{2}-{x}_{1}{\right)}^{2}+\left({y}_{2}-{y}_{1}{\right)}^{2}}$
It is derived from the Pythagorean theorem.

## 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 $\left(1,2\right)$ and $\left(9,8\right)$:
$\begin{array}{rl}& \phantom{=}\sqrt{\left({x}_{2}-{x}_{1}{\right)}^{2}+\left({y}_{2}-{y}_{1}{\right)}^{2}}\\ \\ & =\sqrt{\left(9-1{\right)}^{2}+\left(8-2{\right)}^{2}}\phantom{\rule{1em}{0ex}}\text{Plug in coordinates}\\ \\ & =\sqrt{{8}^{2}+{6}^{2}}\\ \\ & =\sqrt{100}\\ \\ & =10\end{array}$
Notice: we were careful to put the $x$-coordinates together and the $y$-coordinates together and not mix them up.

Problem 1
What is the distance between $\left(4,2\right)$ and $\left(8,5\right)$?

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?
• 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.
• what is the Pythagorean therom
• 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
• 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?
• 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.
• 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.
• Dont understand the graph
• 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.
• Why is the exponent at the bottom?
• This is not an exponent, it just shows the order.
• I find it so funny that the khan comments are so harmonic.
• 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?
• 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