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## High school geometry

### Course: High school geometry>Unit 6

Lesson 1: Distance and midpoints

# Midpoint formula review

Review the midpoint formula and how to apply it to solve problems.

## What is the midpoint formula?

The formula gives the midpoint of the 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 in the coordinate plane:
left parenthesis, start color #1fab54, start fraction, x, start subscript, 1, end subscript, plus, x, start subscript, 2, end subscript, divided by, 2, end fraction, end color #1fab54, comma, start color #e07d10, start fraction, y, start subscript, 1, end subscript, plus, y, start subscript, 2, end subscript, divided by, 2, end fraction, end color #e07d10, right parenthesis
The first quadrant of a coordinate plane with three tick marks on the x axis labeled x one, x one plus x two all divided by two, and x two. There are three tick marks on the y axis labeled y one, y one plus y two all divided by two, and y two. There is a point at x one, y one and another point at x two, y two. A third point is the midpoint of the two others at x one plus x two all divided by two, y one plus y two all divided by two.

## What problems can I solve with the midpoint formula?

Given two points on the plane, you can find their midpoint. For example, let's find the midpoint of left parenthesis, start color #1fab54, 5, end color #1fab54, comma, start color #e07d10, 3, end color #e07d10, right parenthesis and left parenthesis, start color #1fab54, 1, end color #1fab54, comma, start color #e07d10, 7, end color #e07d10, right parenthesis:
\begin{aligned} &\phantom{=}\left(\greenD{\dfrac{x_1+x_2}{2}}, \goldD{\dfrac{y_1+y_2}{2}}\right) \\\\ &=\left(\greenD{\dfrac{5+1}{2}}, \goldD{\dfrac{3+7}{2}}\right)\quad\small\gray{\text{Plug in coordinates}} \\\\ &=(\greenD3, \goldD5) \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 midpoint of left parenthesis, 6, comma, 2, right parenthesis and left parenthesis, 10, comma, 0, right parenthesis?

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

## Want to join the conversation?

• How do you find the endpoint of a line when you are given the other endpoint and the midpoint?
• Great question, and it's definitely a situation we need to be able to solve.

When we are given the Midpoint and an Endpoint, and asked to find the Other Endpoint.

We can still use the concepts from the Midpoint Formula by plugging in the information we do have, because…

we normally find the Midpoint by adding the corresponding coordinates, then dividing by two…

(first endpoint + second endpoint) divided by two = Midpoint

So, we know that…
(x1 + x2)/2 = Midpoint x
and that…
(y1 + y2)/2 = Midpoint y

the given Endpoint (x, y) can be plugged into either the first or second x and y locations because addition is Commutative, (it's rearrangeable), so it doesn't matter which end values are first or second,

and the given Midpoint, (x or y), is placed as equal to the Endpoint addition then division.

Given:
Endpoint (3, 5)
Midpoint (7, 10
)

•Find Other Endpoint
=
(x1 + x2)/2 = Midpoint x
and
(y1 + y2)/2 = Midpoint y
=
(x + 3)/2 = 7
and
(y + 5)/2 = 10

•Let's solve for x first:
(x + 3)/2 = 7
Multiply both sides by 2,
to cancel the division on the left.
=
2((x + 3)/2) = 7 • 2
=
x + 3 = 14
Subtract 3 from both sides,
to isolate x on the left.
=
x + 3 -3 = 14 -3
=
x = 11 ←🥳

•Solve for y the same way…
(y + 5)/2 = 10
=
2((y + 5)/2) = 10(2)
=
y + 5 = 20
=
y = 20 -5
=
y = 15 ←🥳

• Other Endpoint is at:
(11, 15) ←🥳🥳

•Each time the pattern of operations will be the same:
Multiply the Midpoint coordinate by 2, then add or subtract the given Endpoint to isolate variable.

2 • (Midpoint x or y) +/- (endpoint)
=
other endpoint

•Be aware if a given Endpoint coordinate is negative, it will be added rather than subtracted to isolate the variable.

(ㆁωㆁ)I hope this helps someone!
• don't u divied when u do the midpoint
• You are finding the average, so one way is to add the two x coordinates and divide by 2, then do the same for the y coordinates.
• how do i find coordinators if im looking for the endpoint
• Are you saying you know one endpoint and the midpoint? If this is what you are asking, find the vector from the endpoint to the midpoint, the starting at midpoint, go along an equivalent vector. For example, endpoint is at (5,6) and midpoint is at (3,8) so the vector is <3-5, 8-6> or <-2,2>. SO starting at (3,8) along vector <-2,2), you end at (1,10).
• formula for the length of scalene triangle
(1 vote)
• what is length triangle mean?
• it means what is the length of the triangle
• How do you find the values of a and b when given the endpoints (1,a) and (b,-4) and the midpoint (4,-6)?
• Plot two points to find me square in the above graph. Label the points R and S. A square is a quadrilateral with all for size of the same length. After planting the points trace to square by connecting the points
• how do you find the midpoint of 3 coordinates