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.

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.

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.

What's the best way to remember the Distance formula?

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!

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

Am I allowed to use trigonometry to solve these?

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.

Main content

High school geometry

Course: High school geometry > Unit 6

Lesson 1: Distance and midpoints

Distance formula review

Google Classroom

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.

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

Current

What is the distance between left parenthesis, 4, comma, 2, right parenthesis and left parenthesis, 8, comma, 5, right parenthesis?

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

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17017
Posted 6 years ago. Direct link to 17017's post “The soccer field question...”
The soccer field question confuses me - I don't understand how =√52 becomes 2√13!
Any help?
Button navigates to signup pageComment on 17017's post “The soccer field question...”
(35 votes)
Answer
- Ian Pulizzotto
  Posted 6 years ago. Direct link to Ian Pulizzotto's post “To simplify √52, we look ...”
  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.
  Comment on Ian Pulizzotto's post “To simplify √52, we look ...”
  (99 votes)
mari
Posted 7 months ago. Direct link to mari's post “these comments are actual...”
these comments are actually so nice
Button navigates to signup pageButton navigates to signup page
(11 votes)
Answer
kristofer
Posted 6 years ago. Direct link to kristofer's post “what is the Pythagorean t...”
what is the Pythagorean therom
Button navigates to signup pageComment on kristofer's post “what is the Pythagorean t...”
(3 votes)
Answer
- Daniel Ye
  Posted 5 years ago. Direct link to Daniel Ye's post “The pythagorean therom st...”
  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
  Comment on Daniel Ye's post “The pythagorean therom st...”
  (17 votes)
Mateo Bump
Posted 2 years ago. Direct link to Mateo Bump's post “Hello, I Don't understand...”
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?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Jerry Nilsson
  Posted 2 years ago. Direct link to Jerry Nilsson's post “The two points are (𝑥₁, ...”
  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
  Comment on Jerry Nilsson's post “The two points are (𝑥₁, ...”
  (12 votes)
Remedy
Posted 7 months ago. Direct link to Remedy's post “I have a question about t...”
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?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- David Severin
  Posted 7 months ago. Direct link to David Severin's post “anytime you can substitut...”
  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.
  Comment on David Severin's post “anytime you can substitut...”
  (8 votes)
349933
Posted 3 years ago. Direct link to 349933's post “Dont understand the graph”
Dont understand the graph
Button navigates to signup pageComment on 349933's post “Dont understand the graph”
(4 votes)
Answer
- BraedenHill001
  Posted 3 years ago. Direct link to BraedenHill001's post “The graph is showing the ...”
  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.
  Button navigates to signup page
  (2 votes)
Quinn Hays
Posted a year ago. Direct link to Quinn Hays's post “Am I allowed to use trigo...”
Am I allowed to use trigonometry to solve these?
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- David Severin
  Posted a year ago. Direct link to David Severin's post “If you use trig, it will ...”
  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.
  Button navigates to signup page
  (9 votes)
eva stepherson
Posted 3 years ago. Direct link to eva stepherson's post “It all confuses me. But m...”
It all confuses me. But mostly the soccer feild.
Button navigates to signup pageComment on eva stepherson's post “It all confuses me. But m...”
(4 votes)
Answer
792569
Posted 3 years ago. Direct link to 792569's post “how do u solve it if the ...”
how do u solve it if the 2 x's are negetavie
Button navigates to signup pageComment on 792569's post “how do u solve it if the ...”
(4 votes)
Answer
- Monil Mehta
  Posted a year ago. Direct link to Monil Mehta's post “You can still subtract a ...”
  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!
  Comment on Monil Mehta's post “You can still subtract a ...”
  (0 votes)
Brian S
Posted a month ago. Direct link to Brian S's post “What's the best way to re...”
What's the best way to remember the Distance formula?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Ian Pulizzotto
  Posted a month ago. Direct link to Ian Pulizzotto's post “The best way to remember ...”
  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!
  Button navigates to signup page
  (3 votes)