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## 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:
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.

## 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.

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.

## 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
• how do u solve it if the 2 x's are negetavie
• 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)
• 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
• 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.
• how do you use the Pythagorean Theorem
• 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!
• It all confuses me. But mostly the soccer feild.
• find a point on the y axis which is equidistant from the points [5,2] and [-4,3]
• After finding the prime factorization you have to eliminate the 2's since there are 2 of them and put them outside the radical.