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From Wikipedia

Midpoint

The midpoint (also known as class mark in relation to histogram) is the middle point of a line segment. It is equidistant from both endpoints.

Formulas

The formula for determining the midpoint of a segment in the plane, with endpoints (x1) and (x2) is:

\frac{x_1 + x_2}{2}

The formula for determining the midpoint of a segment in the plane, with endpoints (x1, y1) and (x2, y2) is:

\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

The formula for determining the midpoint of a segment in the plane, with endpoints (x1, y1, z1) and (x2, y2z2) is:

\left(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)

More generally, for an n-dimensional space with axes x_1, x_2, x_3, \dots, x_n\,\!, the midpoint of an interval is given by:

\left(\frac{x_{1_1} + x_{1_2}}{2}, \frac{x_{2_1} + x_{2_2}}{2}, \frac{x_{3_1} + x_{3_2}}{2}, \dots , \frac{x_{n_1} + x_{n_2}}{2} \right)

Construction

The midpoint of a line segment can be located by first constructing a lens using circular arcs, then connecting the cusps of the lens. The point where the cusp-connecting line intersects the segment is then the midpoint. It is more challenging to locate the midpoint using only a compass, but it is still possible.


Line segment

In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment is either an edge (of that polygon) if they are adjacent vertices, or otherwise a diagonal. When the end points both lie on a curve such as a circle, a line segment is called a chord (of that curve).

Definition

If V\,\! is a vector space over \mathbb{R} or \mathbb{C}, and L\,\! is a subset of V,\,\! then L\,\! is a line segment if L\,\! can be parameterized as

L = \{ \mathbf{u}+t\mathbf{v} \mid t\in[0,1]\}

for some vectors \mathbf{u}, \mathbf{v} \in V\,\!, in which case the vectors \mathbf{u} and \mathbf{u+v} are called the end points of L.\,\!

Sometimes one needs to distinguish between "open" and "closed" line segments. Then one defines a closed line segment as above, and an open line segment as a subset L\,\! that can be parametrized as

L = \{ \mathbf{u}+t\mathbf{v} \mid t\in(0,1)\}

for some vectors \mathbf{u}, \mathbf{v} \in V\,\!.

An alternative, equivalent, definition is as follows: A (closed) line segment is a convex hull of two points.

Properties

In proofs

In geometry, it is sometimes defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC.

In an axiomatic treatment of Geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else defined in terms of an isometry of a line (used as a coordinate system).

Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets to the analysis of a line segment.



From Yahoo Answers

Question:If you have a line segment what is the formula for finding the midpoint?

Answers:You take the length of the segment and divide it by two. If it is on a coordinate plane, then you average the x and y values, (add both up and divide by two), and take those averages, and they are the coordinates of the midpoint.

Question:Point C(2,1) is the midpoint of line segment AB. a.) Find the coordinates of points A and B if they are both on axes. b.) If point A is located on an axis, but point B is not on an axis, in which quardrant(s) can B be located? i dont know how solve this using the length and midpoint of line segment formulas....

Answers:a.) Put A on the x-axis, so it must have coordinates of (x, 0). Put B on the y-axis, so it must have coordinates (0, y). Midpoint formula, where x(m) is the x-coordinate of the midpoint, and y(m) is the y-coordinate of the midpoint: (x(m), y(m)) = [((x(a) + x(b))/2, ((y(a) + y(b))/2] Plug in what you know: (2, 1) = [(x + 0)/2, (0 + y)/2] So, one coordinate at a time, x-coordinate first: 2 = (x + 0)/2 4 = x + 0 4 = x y-coordinate: 1 = (0 + y)/2 2 = 0 + y 2 = y The points are... A: (4, 0) B: (0, 2) b.) This one doesn't use formulas...it is a "what if" question. C is in Quadrant I. If A is on the x-axis, B could be in Quadrant I or Quadrant II. If A is on the y-axis, B could be in Quadrant I or Quadrant IV. Hope that helps!

Question:A line has a midpoint. Always true, sometimes true, or never true?

Answers:False! A line, by definition, continues off to infinity in both directions. Therefore, it doesn't have a midpoint. If you think of the number line, you might consider 0 to be its midpoint. However, that's not quite accurate. You could just as easily consider 347 to be the midpoint, since for any integer n, you can find (347 + n) and (347 - n) on the number line. By this reasoning, 347 is just as good of a "midpoint candidate" as 0 is. A line segment will always have a midpoint, but a line segment isn't the same thing as a line. Hope that helps!

Question:if i have a line segment where i know the midpoint, slope, and length, how can i find the endpoints? I assume i would need to combine the midpoint formula, slope formula, and distance formula somehow.

Answers:You have the coordinates (x1,y1) of the midpoint, the slope m and the length L. Use the slope formula: m = (y2-y1)/(x2-x1) and the distance formula: d = L/2 = {y2-y1)^2 + (x2-x1)^2} To create two equations in x2 and y2. Now you have two variables and two equations so you can get the values (x2,y2) for the upper end of the line and for the lower end.

From Youtube

Finding distances and midpoints of line segments 8th grade project :Two 8th grade students end of year project on finding the distance and midpoints of line segments