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The Midpoint Formula on the SAT

SAT Math · Geometry & Trigonometry · Updated October 2026

What the midpoint is

The midpoint is the average of the two endpoints. Average the $x$-coordinates, then average the $y$-coordinates. For $(x_1, y_1)$ and $(x_2, y_2)$ that is $\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)$.

When a question gives you the midpoint and one endpoint, think of the midpoint as halfway. The step from the first endpoint to the midpoint repeats once more to reach the other endpoint.

Midpoint questions are often quick points. They also turn up inside circle questions, where the center is the midpoint of a diameter. The mistake I see most is subtracting the coordinates instead of averaging them.

246810246810 A B M

$A = (2, 4)$ and $B = (8, 10)$. The midpoint is $M = \left(\dfrac{2 + 8}{2}, \dfrac{4 + 10}{2}\right) = (5, 7)$. The dashed legs show the run of $6$ and rise of $6$, which the distance formula uses.

What you will see

You seeDo this
Two endpoints, find the midpointAverage the $x$'s and average the $y$'s.
The midpoint and one endpointOther endpoint $= 2 \times$ midpoint $-$ known endpoint, for each coordinate.
The ends of a circle's diameterThe center is their midpoint. The radius is half the distance between them.
A midpoint with unknown lettersWrite one equation for the $x$'s and one for the $y$'s.
A point a quarter of the way alongStart at the first point and add a quarter of the run and a quarter of the rise.

Worked examples

Example 1: a missing endpoint

The midpoint of a segment is $(4, 1)$ and one endpoint is $(1, -3)$. Find the other endpoint.

  1. From $x = 1$ to the midpoint $x = 4$ is a step of $3$. Another step of $3$ reaches $x = 7$.
  2. From $y = -3$ to $y = 1$ is a step of $4$. Another step of $4$ reaches $y = 5$.
  3. The other endpoint is $(7, 5)$. Check: $\left(\dfrac{1 + 7}{2}, \dfrac{-3 + 5}{2}\right) = (4, 1)$.

Example 2: the center and radius of a circle

A diameter of a circle has endpoints $(-2, 6)$ and $(4, -2)$. Find the center and the radius.

  1. Center: $\left(\dfrac{-2 + 4}{2}, \dfrac{6 + (-2)}{2}\right) = (1, 2)$.
  2. Diameter length: $\sqrt{6^2 + 8^2} = 10$.
  3. Radius: $5$.

Example 3: unknown coordinates

The midpoint of $(a, 3)$ and $(9, b)$ is $(5, -1)$. Find $a + b$.

  1. $x$: $\dfrac{a + 9}{2} = 5$, so $a = 1$.
  2. $y$: $\dfrac{3 + b}{2} = -1$, so $b = -5$.
  3. $a + b = -4$.

Mistakes to avoid

Practice

Sketch the points on scrap paper. It makes most of these much easier.

Easy
What is the midpoint of the segment with endpoints $(2, 4)$ and $(8, 10)$?
  • A$(10, 14)$
  • B$(3, 3)$
  • C$(5, 7)$
  • D$(6, 6)$
Show solution
Answer: C. $\left(\dfrac{2 + 8}{2}, \dfrac{4 + 10}{2}\right) = (5, 7)$. Choice A forgets to divide by $2$, and choice B subtracts.
EasyStudent-produced response
What is the $x$-coordinate of the midpoint of the segment with endpoints $(-3, 5)$ and $(7, -1)$?
Enter your answer.
Show solution
Answer: $2$. $\dfrac{-3 + 7}{2} = \dfrac{4}{2} = 2$.
Medium
The midpoint of a segment is $(4, 1)$. One endpoint is $(1, -3)$. What is the other endpoint?
  • A$(2.5, -1)$
  • B$(7, 5)$
  • C$(3, 4)$
  • D$(-2, -7)$
Show solution
Answer: B. Other endpoint $= (2 \cdot 4 - 1, 2 \cdot 1 - (-3)) = (7, 5)$. Choice A is the midpoint of the two given points, and choice D steps backward.
Medium
A diameter of a circle has endpoints $(-2, 6)$ and $(4, -2)$. What are the coordinates of the center of the circle?
  • A$(1, 2)$
  • B$(3, -4)$
  • C$(2, 4)$
  • D$(-3, 4)$
Show solution
Answer: A. The center is the midpoint of a diameter: $\left(\dfrac{-2 + 4}{2}, \dfrac{6 - 2}{2}\right) = (1, 2)$.
MediumStudent-produced response
The midpoint of the segment with endpoints $(a, 3)$ and $(9, b)$ is $(5, -1)$. What is the value of $a + b$?
Enter your answer.
Show solution
Answer: $-4$. $\dfrac{a + 9}{2} = 5$ gives $a = 1$. $\dfrac{3 + b}{2} = -1$ gives $b = -5$. So $a + b = -4$.
MediumStudent-produced response
Point $M$ is the midpoint of segment $AB$. If $A = (-6, 2)$ and $M = (-1, 5)$, what is the $y$-coordinate of $B$?
Enter your answer.
Show solution
Answer: $8$. From $A$ to $M$, $y$ goes up by $3$. Another $3$ reaches $8$. Or: $2 \cdot 5 - 2 = 8$.
HardStudent-produced response
A diameter of a circle has endpoints $(-2, 6)$ and $(4, -2)$. What is the radius of the circle?
Enter your answer.
Show solution
Answer: $5$. The diameter is $\sqrt{(4 - (-2))^2 + (-2 - 6)^2} = \sqrt{36 + 64} = 10$, so the radius is $5$.
HardStudent-produced response
Point $M$ is the midpoint of segment $AB$. If $A = (1, 1)$ and $M = (4, 5)$, what is the length of segment $AB$?
Enter your answer.
Show solution
Answer: $10$. $AM = \sqrt{3^2 + 4^2} = 5$. The midpoint is halfway, so $AB = 2 \cdot 5 = 10$.
Hard
Point $P$ lies on the segment from $(0, 0)$ to $(12, 8)$, one quarter of the way from $(0, 0)$. What are the coordinates of $P$?
  • A$(6, 4)$
  • B$(4, 3)$
  • C$(9, 6)$
  • D$(3, 2)$
Show solution
Answer: D. A quarter of the run is $\dfrac{12}{4} = 3$ and a quarter of the rise is $\dfrac{8}{4} = 2$, so $P = (3, 2)$. Choice A is the midpoint, and choice C is three quarters of the way.
HardStudent-produced response
The midpoint of the segment with endpoints $(2k, 5)$ and $(6, k + 1)$ lies on the line $y = x$. What is the value of $k$?
Enter your answer.
Show solution
Answer: $0$. The midpoint is $\left(k + 3, \dfrac{k + 6}{2}\right)$. On $y = x$, the coordinates are equal: $\dfrac{k + 6}{2} = k + 3$, so $k + 6 = 2k + 6$ and $k = 0$. Check: the endpoints are $(0, 5)$ and $(6, 1)$, with midpoint $(3, 3)$.

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