A segment is a part of a line bounded by two points, the shortest distance between these points. There are several ways to compare geometric figures; the choice of this method often depends not only on the conditions of the problem, but also on the possibilities. We will tell you how to compare segments in this article.
Ways to compare two segments
In geometry, two figures that have the same size and shape are called equal. Comparing the figures makes it possible to tell whether they are the same. One way is overlay. If the figures can be combined by overlapping, they are considered equal.
Comparing figures means determining which of them is longer or shorter. The answer must be definite; it cannot be said that one segment is longer or equal to the second. In mathematics, such an answer is incorrect; it can be equated to the absence of an answer.
Write the comparison result using the greater than, less than and equal signs (>;<; =). Например, длина отрезка АБ - 2 см, а ВГ - 8 см, записываем результат сравнения так: АБ < ВГ или ВГ >AB.
You can compare figures in different ways, the choice of which depends on the capabilities or conditions:
- visual method;
- measuring;
- comparison by overlay;
- grid comparison.
It's best if they vary in length visually, and just by looking at them you can tell which is longer. But this doesn't always happen.
Length measurement
The easiest way is to measure. To do this, you can use a ruler; simply by measuring the length of the segment, we will understand which one is longer. If there is no ruler, but they are drawn on a sheet of paper in a square, you can count the squares to measure their lengths . There are two cells in one centimeter. This is a method of comparison by measuring lengths, but there is also a method of comparison by superimposition.
Overlapping
How AB and VG are combined:
- You need to combine the end A of one of them with the end B of the other, if the other ends of these segments - B and D - also coincide, then they are equal, which is written using the equal sign.
- If not, then one of them is longer than the other, and this is also written using the mathematical signs greater than or less than (> or<).
It happens that when one segment is superimposed on another, exactly half of one of them will be combined with the other. The point that divides it into two equal parts is called the midpoint. And if we have a midpoint B, then AB=BB.
In approximately the same way, not only straight lines, but also other geometric shapes, as well as angles, are compared by superposition.
You can make a “ruler” from a strip of paper, and such a ruler does not need to be lined; it is enough to mark the beginning and end of one of the segments on it. Then you apply a makeshift ruler to the second one, aligning its beginning with the first mark and comparing the location of the second mark in relation to its end. In this way, you can also compare fairly large figures, for example, the distance between fence posts, but it is better to use a rope rather than a paper strip.
Two segments are said to be equal, if they can be combined by superposition. If you can put them next to each other, just see which one is longer. But this cannot always be done.
If you have a compass at hand, place one leg of the compass at the beginning and the other at the end of the first segment. Then, without moving the legs of the compass, install one of them at the beginning of the second and see if the second leg of the compass is at the point indicating the end - they are equal. If the second leg is on the straightest line, the first segment is smaller, if behind it, the first segment is larger.
Grid comparison
Let's assume that we have two segments whose coordinates we know - a (X1, Y1; X2, Y2) and b (X3, Y3; X4, Y4).
The first thing to do is give the coordinates numerical values:
- Length, a - Da = √((X1 - X2) ² + (Y1 - Y2) ²);
- Length b - Db = √((X3 - X4)² + (Y3 - Y4)²).
Let X1 = -7, Y1 = 4, X2 = 3, Y2 = -4, X3 = -3, Y3 = -5, X4 = 0, Y4 = -3. We get:
Da = √ ((-7 - 3)² + (4 - (-4))²) = √ (-10² + 8²) = √ 100 + 64 = √ 164
Db = √ ((-3 - 0) ² + (-5 - (-3)) ²) = √ (-3 ² + (-8) ²) = √ (9+ 64) = √ 73
√ 164 > √ 73, which means Da > Db.
You can also compare segments located in a three-dimensional coordinate system; you need to take into account not two, but three coordinates of each of them.
Examples
Let's consider a comparison using the superposition method. We have two segments - AB and VG.
To find out whether they are equal or not, we simply apply them to each other so that their “beginnings” are at the same point, that is, we combine points A and B.
If we see that AB is part of VG, it means that it is smaller, that is, AB< ВГ, а если при наложении оба конца отрезков совмещаются - значит, они равны.
Now let's look at comparing segments by measuring. Using a ruler we calculate the length each segment. For example, length AB = 2 cm, and CD = 8 cm. 8>2, which means CD>AB, that is, segment CD is longer than AB.
A person constantly has to compare objects by size. To combine parts, make a pattern exactly to size, or buy furniture that will definitely fit in your apartment, you need to know whether the parameters of different items correspond to each other. This means that you need to compare the length of two segments.
You will need
- - specified segments;
- - ruler and other measuring instruments;
- - compass.
Instructions
- Remember what a segment is. This is a section of a straight line, bounded on both sides by points. Let's say you are given 2 segments located on the same plane parallel to each other and at the same time the perpendicular dropped from the starting point of one of them will be exactly at the beginning of the other. In this case, use the alignment method. From the end point of the first segment, lower another perpendicular towards the second. If this new line intersects the second segment, this means that the first is shorter than the second, and the second is longer than the first.
- Much more often we have to deal with comparisons of non-parallel segments. In this case, use a measuring compass. Spread its legs to a distance corresponding to the length of one of the segments. Then place one leg at the starting point of the second segment. The second one should be either on the segment or on its continuation. This method is used when you do not need to know the length of both segments, but just need to determine which of them is shorter or longer.
- To compare segments that are not in the same plane, use the standard method. The simplest standard is an ordinary school ruler with divisions. But other measuring instruments can also be used in this capacity. In order to compare two segments drawn on a sheet, attach the zero hole of the ruler to the starting point of one of them. Measure the length of the first segment, and then measure the second in exactly the same way. In this case, you first find the numerical value of the length of the first segment, then the second, and finally compare these values.
- Any sufficiently long object can be used as a temporary reference. This could be, for example, a rope or a batten. This measurement method is used when it is necessary to compare segments, but the numerical value does not play a big role. For example, you need to determine whether a closet will fit between a sofa and a table or not. Tie a knot in the rope. Mark a point on the wall or baseboard near the table or sofa. Lay the rope strictly horizontally and tie a second knot. In the store you will only need to measure the cabinet according to the width of this rope.
How to compare segments?
What does it mean to compare two segments? This means comparing their lengths, determining which one is longer (or shorter). If you have a ruler at hand, there is nothing simpler: use it to measure the lengths of both segments, and it will immediately become clear which is longer. Below we will tell you what to do if there is no ruler near you.
How to compare two line segments without a ruler
If the segments are drawn by cells, you can count the cells. However, this is not always the case. If there are no cells, you can use a compass. First you need to install the compass opening at the ends of one segment, and then, without moving its legs, install the needle at the end of another segment and see whether the compass opening is wider than the second segment, or narrower.
If you don’t have a compass, you can make something like a ruler from a strip of paper. It is not necessary to draw divisions on it; it is enough to mark the beginning and end of one segment, then combine one mark with the beginning of the second segment and compare.
This way you can even compare segments drawn on the ground, for example, in order to designate places for posts for a bench at equal distances from the wall of the house. Only in this case you will need to use not a strip of paper, but a board or rope.
How to compare two segments in a coordinate grid
To compare segments, you need to know their lengths. In the article, we explained how to find the length of a segment if its coordinates on a plane or in space are indicated. Let's take segments on the plane with coordinates: segment a = (x 1,y 1;x 2,y 2) and segment b = (x 3,y 3;x 4,y 4).
Of course, it’s already clear that the second segment is shorter than the first, but in mathematics “it’s visible” doesn’t count, you have to prove it. Therefore, we will write a formula for calculating the lengths of segments and give the coordinates numerical values. After this, you can easily explain how to compare two segments.
- Length of segment a d1 = √((x 1 - x 2)² + (y 1 - y 2)²)
- Length of segment b d2 = √((x 3 - x 4)² + (y 3 - y 4)²)
Let x 1 = -6, y 1 = 5; x 2 = 4, y 2 = -3; x 3 = -2, y 3 = -4; x 4 = 1, y 4 = -2. Means: 
- d1 = √((x 1 - x 2)² + (y 1 - y 2)²) = d1 = √(((-6) - 4)² + (5 - (-3))²) = √( (-10)² + 8²) = √164
- d2 = √((x 3 - x 4)² + (y 3 - y 4)²) = √(((-2) - 1)² + ((-4) - (-2))²) = √ ((-3)² + 2²) = √13
- √164 > √13, which means d1 > d2.
Similarly, you can compare segments in three-dimensional coordinates, only then you will also need to take into account third coordinates: segment a = (x 1,y 1,z 1;x 2,y 2,z 2) and segment b = (x 3,y 3 ,z 3;x 4,y 4,z 4).
The formulas are similar to what we wrote for a coordinate grid on a plane:
- Length of segment a d1 = √((x 1 - x 2)² + (y 1 - y 2)² + (z 1 - z 2)²)
- Length of segment b d2 = √((x 3 - x 4)² + (y 3 - y 4)² + (z 3 - z 4)²)
Let x 1 = -6, y 1 = 5, z 1 = 1; x 2 = 4, y 2 = -3, z 2 = 2; x 3 = -2, y 3 = -4, z 3 = 3; x 4 = 1, y 4 = -2, z 4 = -11.
- d1 = √((x 1 - x 2)² + (y 1 - y 2)² + (z 1 - z 2)² = √(((-6) - 4)² + (5 - (-3) )² + (1 - 2)²) = √((-10)² + 8² + (-1)²) = √165
- d2 = √((x 3 - x 4)² + (y 3 - y 4)² + (z 3 - z 4)²) = √(((-2) - 1)² + ((-4) - (-2))² + (3 - (-11))²) = √((-3)² + 2² + 14²) = √(9 + 4 + 196) = √209
- √209 > √165
This means that in this case the second segment turned out to be larger than the first.
To compare two segments means to determine the length of which of them is greater or less than the other. In the real world, many of us perform such operations without noticing it. We compare the lengths of roads on the map in order to choose a shorter path, we determine the taller of the brothers by measuring and comparing their height, and at a line or in a factory, comparisons of lengths of similar values are used all the time. Our task is to be able to build a mathematical model for any problem, to be able to solve it correctly. You can also compare two segments by eye or using available tools. Let's say what is longer: a match or a ballpoint pen cap? By measuring the length of the match with a compass and applying it to the cap, we can immediately get the answer to the question.
But how to compare two segments if their length is indistinguishable by eye? If it is not possible to use available means, and we are given only the coordinates of the segment? In the case of one-dimensional space, you can compare two segments by finding their lengths. On a straight line, the length of a segment is the difference in the coordinate values of its ends, taken with a plus sign. For example: given a segment AB with coordinates A(2), B(3) and a segment CD with coordinates C(5,1) and D(6). Determine which of the segments is longer. The length AB will be equal to 3-2 = 1, and the length CD will be equal to 6-5.1 = 0.9. It follows from this that segment AB is greater than CD. Let's consider another problem. The coordinates of the segment KL are given: 0 and 4, respectively. Also given are the coordinates of the beginning of the segment MN M(-3) and the coordinate of the middle of this segment (-1). Compare the lengths of the segments KL and MN.
To solve such a problem, you need to know how to find the coordinates of the midpoint of a segment. The coordinate of the middle of a segment is the arithmetic mean of the coordinates of its ends. For our problem, it turns out that the coordinate M(-3) plus the unknown coordinate N(x) when divided in half will give -1. Let's compose and solve the equation. (-3+x) /2 = -1. Let's multiply both sides by -2: -3+x= -2. Let's move -3 to the right side of the equation, changing the sign: x=1. We find that the coordinate N is equal to 1. Find the length of the segment MN: 1-(-3) =1+3=4. Similarly, the length KL = 4-0 = 4. As you can see, the lengths of the segments are the same, therefore the segments are equal.
For geometric problems, it is often important to know the name of the segment connecting certain two points. Sometimes this will help you avoid solving the problem in general form and apply the theorem and a simplified method for solving the problem. However, let’s solve the problem where the general formula for finding the length of a segment through the coordinates of its ends is used. For a plane, the length of a segment is equal to the root of the sum of the squares of the differences between the corresponding coordinates of its ends. This formula is a generalization for one-dimensional space, which. in turn, is a special case of the formula for three-dimensional and so on. Knowing how to use such formulas, you can both find the length of a segment on a plane and in space. Let's move directly to the task.
Task. Point C with coordinates (-3;2) is the common beginning of the segments NE and CA. Point A has coordinates (0;0), and point B has coordinates (1;4). Compare segments NE and SA. Solution. Let's calculate the length of the segment SA using the formula described above: the root of -3-0 = -3 squared, this value is equal to 9.2-0 = 2, squaring two we get 4. The sum of these squared differences is 13, therefore the length of SA is equal to the root of 13. Applying similar arithmetic operations to find the length of SV, we find that the length of this segment is -3-1 = -4. -4*-4=6.2-4 = -2. -2*-2 = 4.6+4 = 20, hence the length of the segment CB is equal to the root of 20. The root of 20 is greater than the root of 13, therefore the segment CB is greater than the segment CA. The problem is solved.
