A midpoint is the point exactly halfway between the two endpoints of a line segment, so it splits the segment into two equal parts. On the coordinate plane, find it by averaging the coordinates: the midpoint of (x1, y1) and (x2, y2) is (x1 + x22, y1 + y22).
In Spanish
punto medio
Also called
midpoint of a line segment, midpoint formula, halfway point
2 is halfway between −2 and 6, and 3 is halfway between 1 and 5, so the midpoint is (2, 3).
Formal definition and notation
The midpoint of AB is the point M on AB with AM = MB. With A = (x1, y1) and B = (x2, y2), M = (x1 + x22, y1 + y22). It is the point that partitions AB in the ratio 1 : 1, and every segment has exactly one midpoint.
Both mean “middle”. The median is the middle value of a data set; the midpoint is the middle point of a segment. The midpoint of 2 and 10 is 6, and the median of 2, 3, 10 is 3.
Midpointvs.Distance
The distance formula gives the length of a segment, a single number; the midpoint formula gives a point, an ordered pair. From (0, 0) to (6, 8): distance 10, midpoint (3, 4).
The segment is the whole piece between two endpoints; the midpoint is the single point in its exact center.
Quick check: 3 questions
Question 1 of 3
What is the midpoint of (−6, 2) and (4, 8)?
Show answer and explanation
Answer: A.−6 + 42 = −1 and 2 + 82 = 5, so (−1, 5). (−5, 3) and (5, 3) halve the differences instead, and (−2, 10) forgets to divide by 2.
Question 2 of 3
On a number line, what is the midpoint of 3.5 and 12.5?
Show answer and explanation
Answer: C.3.5 + 12.52 = 162 = 8. 4.5 is half the distance, 9 is the distance, and 16 is the sum before dividing.
Question 3 of 3
M(2, 0) is the midpoint of PQ and P = (−3, 4). What is Q?
Show answer and explanation
Answer: C.Q = (2 × 2 − (−3), 2 × 0 − 4) = (7, −4). (−0.5, 2) averages M and P, and (5, −4) forgets the double.
Midpoint: frequently asked questions
What is the midpoint formula?
For endpoints (x1, y1) and (x2, y2), the midpoint is (x1 + x22, y1 + y22). In words: average the x-coordinates and average the y-coordinates. For (1, 4) and (7, 10), the midpoint is (4, 7). It works in every quadrant, with negative coordinates too.
Can a midpoint have decimal or fraction coordinates?
Yes. When the sum of two coordinates is odd, dividing by 2 gives a half. The midpoint of (1, 2) and (4, 7) is (2.5, 4.5). That is a real point on the grid, halfway between gridlines, and you should not round it unless the problem says to.
How do you find an endpoint from the midpoint?
Double the midpoint and subtract the known endpoint, one coordinate at a time: x2 = 2xM − x1 and y2 = 2yM − y1. Or think in steps: whatever move takes you from the known endpoint to the midpoint, make the same move again to reach the other endpoint.
Line SegmentA line segment is a straight part of a line with two endpoints. It starts at one point, stops at another, and includes every point between them. A segment with endpoints A and B is written AB. Unlike a line, which goes on forever, a segment has a definite length that you can measure.Geometry · Grade 4, Grade 5, Grade 6RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3AverageAn average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.Statistics · Grade 5, Grade 6, Grade 7SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1Congruent FiguresCongruent figures have exactly the same size and shape. One can be moved onto the other by sliding, flipping or turning, so every pair of matching sides and angles is equal. Two squares with side 5 are congruent; a 3-4-5 triangle and a 6-8-10 triangle are not, because their sizes differ. The symbol is ≅.Geometry · Grade 7, Grade 8Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Midpoint FormulaThe midpoint formula gives the point halfway between (x1, y1) and (x2, y2): M = (x1 + x22, y1 + y22). You average the two x-coordinates and, separately, the two y-coordinates. For (2, 1) and (10, 7), the midpoint is (6, 4).Geometry · Grade 8, Algebra 1, GeometryMedianThe median is the middle value when data are listed in order. If there is an even number of values, the median is the average of the two middle values. Half the data lie at or below the median and half at or above it, and outliers barely move it.Statistics · Grade 5, Grade 6, Grade 7DataData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3, Grade 6Distance FormulaThe distance formula finds the straight-line length between two points on the coordinate plane: d = √(x2 − x1)2 + (y2 − y1)2. It is the Pythagorean theorem in disguise: the horizontal and vertical differences are the legs of a right triangle, and the distance is the hypotenuse.Geometry · Grade 8, Algebra 1, GeometryOrdered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6, Grade 7QuadrantA quadrant is one of the four regions that the x-axis and y-axis split the coordinate plane into. They are numbered I, II, III and IV counterclockwise, starting at the top right. The signs of a point’s coordinates tell its quadrant: (+, +) is I, (−, +) is II, (−, −) is III and (+, −) is IV.Geometry · Grade 5, Grade 6, Grade 7Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7MeanThe mean is the sum of the values divided by the number of values. It is the arithmetic average: add every value and divide by how many there are. For 3, 7, 5 and 9, the mean is 24 ÷ 4 = 6. The mean is the balance point of the data, so one outlier can pull it far.Statistics · Grade 5, Grade 6, Grade 7Equation of a CircleThe equation of a circle with center (h, k) and radius r is (x − h)2 + (y − k)2 = r2. It says that every point (x, y) on the circle is exactly r units from the center. A circle centered at the origin has the simpler equation x2 + y2 = r2.Geometry · Geometry, Algebra 2, Precalculus