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## points

Points and lines[1]

The simplest objects in Geometry are Points and Straight Lines

point represents position and has no length, no width, no height, no thickness. straight line has length only; it has no width, no height, no thickness.

The Points and Straight Lines have some properties. Some of these properties are inter-related. For examples,

P1. For every two points in plane, there is a straight line passing through them, and such a line is unique.

P2. For any two points the shortest line between them is the straight line which passes through these points. The two properties P1 and P2 listed above are postulates. Postulates are properties that we consider and accept as granted, without the proof. We consider postulates as valid properties based on everyday people's experience. In Geometry, such postulates are the base to prove other properties of figures.

Two theorems below are examples of deriving properties of figures.

Theorem 1 If two straight lines have two common points, then these straight lines coincide.

## Proof 1

This follows from the postulate P1 above, because only one straight line exists passing through two points.

## Theorem 2

In any triangle, the sum of two sides is longer than the third side.

## Proof 2

The third side of the triangle is the straight line segment connecting two points (the triangle vertices). In accordance to the postulate P2 above,  it is the shortest distance between these points, therefore, it is shorter than the sum of other two sides of the triangle.

We used the expressions "straight line segment", "the length of the segment". I should explain what exactly they mean.

 Figure 1 shows the straight line and two points A and Bin it. These two points define the straight line segment,     or simply straight segment, which is the set of all points of the straight line Figure 1. The straight line segment AB that are located in between  points A and B. It is natural to call this straight segment as AB using the names of its endpoints. Note that  endpoints A and B are included to the segment AB also.
 Figure 2 shows the straight line and three points A, Band C in it.  P3. For every three points in the straight line, one of them is located in between two figure 2 the straight line segment ABC others.              This statement is the postulate too. In Figure 2 the point B lies between points A and C. You have three straight line segments AC, CB and AB in this Figure. The point B divides the segment AB in  two segments, AC and CB. Segments AC and CB are adjacent and, taken together, form the segment AB.

We can compare any two straight line segments in the plane by moving them as rigid bodies, over-posing their  starting points and aligning them in one direction along some straight line.

## Definition 1

Two segments are called congruent if they can be laid one onto other so that their endpoints coincide. Note that if one straight segment is congruent to the second one, and the second segment is congruent to the third one, then the first straight segment is  congruent to the third one.

If two straight line segments are congruent, they have the same length. The opposite statement is true also: if two straight line segments have the same length, they are congruent.

If one straight segment has the same length as the second one, and the second segment has the same length as the third one, then the first straight segment  has the same length as the third one.

## Definition 2

The distance between two points in the plane is the length of the straight segment connecting these two points. If in the straight line, the two straight segments have the common starting point A and the endpointB of the first segment is located in between  the endpoints A and C of the second segment (as it is shown in Figure 2), we say that the second segment is longer than the first one; the first segment  is shorter than the second one.

It is possible to measure quantitatively the length of the straight line segment. If the straight segment is the part of the number line, then the length of the straight segment is equal to the difference of the numbers  that correspond to the segment endpoints. More exactly, the length of the straight segment is equal to the absolute value of this difference.

There are special tools to measure the length of the straight segment - rulers, for example. When you measure the length of the straight line segment by applying the ruler, you actually use the ruler as the material model of the number line.

There are different units for length measuring. People use feet (ft), inches (in), meters (m), centimeters (cm), millimeters (mm), kilometers (km), miles ... .

1 foot = 12 inches, 1 inch = 1/12 foot, 1 cm = 0.01 m, 1 mm = 0.001 m, 1 cm = 10 mm, 1 m = 1000 mm, 1 inch = 2.54 cm, 1 ft = 30.48 cm, 1 mile = 1609.344 meters.

The next statement is the postulate too.

P4. If the straight line segment is divided into the two lesser segment by the internal point, then the length            of the entire segment is equal to the sum of the lengths of the smaller segments

You can calculate the sum of the lengths of two or more straight line segments in the plane that not necessary lie in one straight line.  For example, the perimeter of the triangle is equal to the sum of the lengths of its three sides.  The perimeter of a quadrilateral is equal to the sum of the lengths of its four sides.  In general, the perimeter of a polygon is equal to the sum of the lengths of all its sides. The length of a polygon line is equal to the sum of the lengths of all its sides, no matter if this polygon line is closed or not.

## Problem 1

If in Figure 2 the length of the straight segment AB is equal to 2 cm and the length of the straight segment BC is equal to 1 cm,  find the length of the straight segment AC

## Solution

The length of the straight segment AC is equal to the sum of the straight segment AB and the length of the straight segment BC that is 2 cm + 1 cm = 3 cm.

The length of the straight segment AC is equal to 3 cm.

## Problem 2

If in Figure 2 the length of the straight segment AC is equal to 3.1 cm and the length of the straight segment BC is equal to 0.9 cm,  find the length of the straight segment AB

## Solution

The length of the straight segment AC is equal to the sum of the straight segment AB and the length of the straight segment BC hence, the length of the straight segment AB is equal to  3.1 cm - 0.9 cm =2.1 cm.

The length of the straight segment AB is equal to 2.1 cm.

## Problem 3

Find the perimeter of the square if its side length is 2.5 cm.

## Solution 3

The perimeter of the square is equal to the sum of the lengths of its four sides. Since the sides of the square have the same length, the perimeter of the square is four times of its side length, that is 4*2.5 cm = 10 cm.