prove that every every line segment has one and only one end point . use euclids geometry to prove
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To prove that every line segment has one and only one endpoint using Euclid's geometry, we will refer to Euclid's five postulates and some definitions.
1. Euclid's First Postulate: A straight line segment can be drawn joining any two points.
2. Euclid's Second Postulate: Any straight line segment can be extended indefinitely into a straight line.
3. Euclid's Third Postulate: Given any line segment, a circle can be drawn having the segment as the radius and one endpoint as the center.
4. Euclid's Fourth Postulate: All right angles are congruent.
5. Euclid's Fifth Postulate (the parallel postulate): If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then those lines, if extended indefinitely, will intersect on that side on which the sum of the interior angles is less than two right angles.
Now let's proceed with the proof:
1. Assume we have a line segment AB. By the first postulate, we can draw a straight line segment joining its two endpoints.
2. According to Euclid's third postulate, we can draw a circle with one endpoint (let's say A) as the center, and AB as the radius.
3. The circle intersects the line segment AB at endpoint B and continues indefinitely in both directions. The circle cannot intersect AB at any other point since they don't have the same distance to the center (A).
4. Now, let's assume that there is another point C on the line segment AB apart from A and B, which is also an endpoint of AB. By definition, the line AC and BC would still be part of the line segment AB.
5. By Euclid's fifth postulate (the parallel postulate), if AC and BC are part of the same line segment AB and are extended indefinitely, they should intersect at some point.
6. However, this contradicts step 3 since the circle intersects the line segment only at endpoint B, and there is no other point of intersection. Thus, point C cannot exist on the line segment AB.
7. Therefore, we conclude that a line segment AB has only one endpoint, which is point B.
By using the five postulates and definitions from Euclid's geometry, we have shown that every line segment has one and only one endpoint.
it's false. A line segment is by definition finite in extent. So, it has two end points.
Since a line segment is non-directional, it makes no sense to call one end point a starting point.
A ray, on the other hand, has one end point, and extends indefinitely from it in a given direction.
Let us consider, a line segment AB.
Assume that it has two midpoints say C and D
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Recall that the midpoint of a line segment divides it into two equal parts
That is AC = BC and AD = DB
Since C is midpoint of AB, we have A, C and B are collinear
∴ AC + BC = AB → (1)
Similarly, we get AD + DB = AB → (2)
From (1) and (2), we get
AC + BC = AD + DB
2 AC = 2AD
∴ AC = AD
This is a contradiction unless C and D coincide.
Therefore our assumption that a line segment AB has two midpoints is incorrect.
Thus every line segment has one and only one midpoint.