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Learn it your way
A short class with Sarah and her students. The other ways are under More.
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On a number line, what number is exactly halfway between 2 and 6?
Now take the points A(2, 3) and B(6, 7). How would you find the point halfway between them?
Average the x's to get 4 and average the y's to get 5. So the midpoint is (4, 5).
How can we check it really is halfway?
Use the distance formula. A to M and M to B are both √8, so it's the same distance from each end.
M(3, 1) is the midpoint of A(−1, 4) and B. Where is B?
Set up (−1 + x)/2 = 3, so −1 + x = 6 and x = 7.
And (4 + y)/2 = 1, so 4 + y = 2 and y = −2. B is (7, −2).
Is there a quicker way to see it?
From A to M you go right 4 and down 3. Do the same again from M and you reach (7, −2).
Exactly. M is halfway, so the second half of the journey matches the first.
The ends of a diameter of a circle are (−3, 2) and (5, 8). Where is the centre?
The centre is halfway along the diameter, so it's the midpoint: (1, 5).
Now ABCD has A(0, 0), B(4, 1), C(6, 4) and D(2, 3). Is it a parallelogram?
The midpoint of AC is (3, 2), and the midpoint of BD is also (3, 2).
So the diagonals bisect each other, which means it's a parallelogram.
Well reasoned. One calculation, twice, gave a proof.
M = ((x₁ + x₂)/2, (y₁ + y₂)/2), found by averaging each coordinate.
half of the horizontal change and half of the vertical change take you to the midpoint.
if M and A are known, B = (2 × M's x − A's x, 2 × M's y − A's y).
the midpoint of a diameter is the centre of the circle.
if both diagonals of a quadrilateral have the same midpoint, it is a parallelogram.
A(2, 3) and B(6, 7): M = ((2 + 6)/2, (3 + 7)/2) = (4, 5).
P(−5, 4) and Q(3, −2): M = ((−5 + 3)/2, (4 − 2)/2) = (−1, 1).
M(3, 1) is the midpoint of A(−1, 4) and B: (−1 + x)/2 = 3 and (4 + y)/2 = 1, so B = (7, −2).
A diameter joins (−3, 2) and (5, 8): the centre is (1, 5) and the radius is half of √(8² + 6²) = 10, which is 5.
Add the x's, split in two; add the y's, halve them too; the point you find is in the middle, solving the midpoint riddle.
Two friends at (2, 3) and (6, 7) agree to meet halfway. Each walks half of the east-west gap and half of the north-south gap, and they bump into each other at (4, 5).
"Midpoint means mean" - the midpoint's coordinates are the means of the endpoints' coordinates.
A.H.A. - Add, Halve, Assemble the coordinates.
Term
State the midpoint formula.
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Term
Find the midpoint of (2, 3) and (6, 7).
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Term
Find the midpoint of (2, −1) and (4, 3).
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M(3, 1) is the midpoint of A(−1, 4) and B. Find B.
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How can midpoints show a quadrilateral is a parallelogram?
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The point that divides a line segment into two equal parts.
M = ((x₁ + x₂)/2, (y₁ + y₂)/2), the averages of the endpoints' coordinates.
One of the two points at the ends of a line segment.
The part of a line between two endpoints.
A chord through the centre of a circle; its midpoint is the centre.
To cut into two equal parts.
The line through the midpoint of a segment at right angles to it; every point on it is equidistant from the endpoints.
The midpoint of the segment joining A(x₁, y₁) and B(x₂, y₂) is M = ((x₁ + x₂)/2, (y₁ + y₂)/2): you average the x-coordinates and average the y-coordinates. For A(2, 3) and B(6, 7) the midpoint is (4, 5).
The midpoint of a line segment is the point exactly halfway between its endpoints, and its coordinates are the averages of the endpoints' coordinates. This lesson explains why averaging works, works through the example A(2, 3) and B(6, 7), whose midpoint is (4, 5), and practises with negative and fractional coordinates. It then turns the formula around to find a missing endpoint, uses it to find the centre and radius of a circle from a diameter, to prove that a quadrilateral is a parallelogram, and to write the equation of a perpendicular bisector.
The formula. For the segment joining A(x₁, y₁) and B(x₂, y₂), the midpoint is
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Each coordinate of M is the mean of the matching coordinates of A and B.
Why averaging works. Going from A to B, the x-coordinate changes by x₂ − x₁ and the y-coordinate by y₂ − y₁. Halfway along the segment, exactly half of each change has happened, so M has x-coordinate x₁ + ½(x₂ − x₁) = (x₁ + x₂)/2, and similarly for y. On a number line, the point halfway between 2 and 6 is 4, the average; the midpoint formula does this separately for x and for y.
Example: A(2, 3) and B(6, 7).
More practice.
Working backwards: a missing endpoint. M(3, 1) is the midpoint of A(−1, 4) and an unknown point B(x, y). Then (−1 + x)/2 = 3, so x = 7, and (4 + y)/2 = 1, so y = −2, giving B(7, −2). A quick way to see it: the step from A to M is +4 in x and −3 in y, and the same step again from M lands on B.
Applications.
Common mistakes. Subtracting the coordinates instead of adding them, dividing only one coordinate by 2, and mixing an x-value with a y-value.