Measurement of Segments (Level 1) | Measuring Segments, Congruent Segments

Measurement of Segments (Level 1) | Measuring Segments, Congruent Segments

Geometry was initially used to solve problems involving measurements on the earth. The word Geometry comes from the Greek word “geo” which means earth and “metry” which means measure. So this is the mathematics that allows us to carryout measurements on the earth.

Although the earth is essentially spherical in shape, when we view the earth in small regions it appears to be flat, this is why in many practical situations where measurements are involved, the earth is assumed to be flat and considered to be part of a plane.

You can think of a plane as a sheet of paper of points with no thickness, stretched tightly, and extending infinitely in all directions. Geometry that involves measurements in a plane is referred to as plane geometry and in this course the majority of the topics that we will cover will allow us to solve 2 dimensional planer geometry problems, later on in this course we will use our new planer geometry knowledge and solve simple 3 dimensional geometry problems. The field in geometry that deals with 3 dimensional figures is known as solid geometry. In the following series of posts we will learn how to measure line segments.

In the previous post we defined a line segment also referred to as simply a segment as a straight line that is made up of points that has a definite beginning and end, these two points are referred to as the end points of the segment. Because segments have a definite beginning and an end they can be measured and have a unique length.

We measure segments by using instruments such as a rulers and meter sticks. We may use any convenient length as a unit of measure. Common units include inches, feet, yards, millimeters, centimeters, and meters. For example segment AB measures 1 inch and segment CD measures 5 centimeters.

In geometry the way we denote the measurement of a segment is by writing the end points of the segment for example the measurement of segment AB is 1 inch and the measurement of segment CD is 5 centimeters. Notice the distinction between denoting a line segment (a geometric figure) and denoting the measurement or length of the line segment which represents a numerical value. If you want to identify the line segment we draw a horizontal segment also known as a bar on top of the letters, if we want to denote the measurement or length of the line segment then we only denote the endpoints and do not draw a horizontal segment or bar on top of the letters.

We can also use a number line to find the measurement of a segment. The real number associated with a point on a line is called the coordinate of that point. For example the following line segment is formed by point P and point Q with point P located at the coordinate negative 4 and point Q located at the coordinate positive 2. Usually, only integers are labeled in a number line because it is impossible to label all the real numbers like fractions or decimals since there are infinite real numbers.

The measurement of this line segment can be determined by measuring how far apart the end points are from one another. In other words we measure the distance between the points. In this case the measurement of segment PQ is 6. We can define the distance between point A and point B as the nonnegative difference between their coordinates for example segment PQ can be found by subtracting the coordinates of the line segment in this case we can subtract negative 4 from positive 2 or subtract positive 2 from negative 4 since we want the numbers to be nonnegative we take the absolute value of the result. In general if a line segment has one end point located at the coordinate A and the second end point located at the coordinate B then the length of the line segment is given by the absolute value of A minus B or the absolute value of B minus A.

Notice that we are not assigning a unit such as inches or centimeters to this length, when the unit is not explicitly stated we can either leave our answer as a number such as 6 or we can say that the line segment measures 6 units in length. Now let’s talk about congruent segments.

In geometry two objects that have the same size and shape are called congruent. For many geometry figures we can give a more precise definition of what it means to be congruent. Later on in this course we will define congruent angles, congruent triangles, and congruent circles. Let’s first define congruent segments.

Congruent segments are segments that have the same lengths in other words the segments have equal measurements. In the figures shown segments AB, CD and EF are congruent. To indicate that segment AB and segment CD have equal lengths or to denote that the distance from point A to point B equals the distance from point C to point D, we write AB equals CD without using the line segment or bar symbol on top of the letters. To indicate that segment AB and segment CD are congruent we use the following symbol. This symbol uses a tilde above the equal sign and is read as: “segment AB is congruent to segment CD”. This statement can be used interchangeably meaning that if segment AB is congruent to segment CD then segment CD is congruent to segment AB. Remember to use equality for numbers and congruence for geometric figures.

We can also use identical tick marks to indicate congruent segments in a diagram or figure, for example in the following figures a single tick mark is used to show that these two line segments are congruent. If multiple segments need to be labeled as congruent then we can use double or triple tick marks, for example double tick marks are used to show that these line segments are congruent and triple tick marks are used to show that these line segments are congruent.

Alright and this is the basics of measuring line segments. We can find the length of a line segment by using a ruler with an appropriate unit or we can use a number line and find the absolute difference of the segments endpoint coordinates. We can also identify and label congruent segments by using tick marks in a diagram or by using the congruence symbol which tells us that two line segments have the same length. In our next post we will go over simple examples that makes use of the concepts learned so far.