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## Meaning of linear measure and precision

**Lines** join two points and extend to infinity. **Line segments** are parts of lines that join two points together. This means that the 'lines' we typically see are geometrically defined as line segments.

This figure provides an example of a line segment as it has two endpoints and doesn't extend infinitely. As it joins the points A and B, it can be referred to as AB.

**Linear measure** is the measurement of the length of a line segment. Length is a measure of distance; so, finding the length of the line segment that joins two points in space would give us the distance between the two points. How do we measure length? We use established units that define a specific amount of length. The unit of one inch always measures the same amount of distance; so, we can measure the length of line segments by how many inches fit into its length.

Above is a tool which is used for linear measure. It is known as a ruler and is a rectangular instrument which has intervals of certain units of length written on it. On this ruler, we can see the use of units, mm (millimeters), which is a metric unit. The numbers on the ruler provide the amount of units of length of the measured object, as measured from the indicated zero.

How do we use a ruler? Line the ruler up with the line segment you want to measure, then place one endpoint of the line segment you want to measure on the zero of the ruler, and note the point on the ruler where the other endpoint of the line segment is. This is the length of the line segment.

The **precision** of a measurement is based on the tool used. Each tool has a certain increment in units which we use to measure the lengths of line segments. However, if the end point of a line falls between two of the increments, we don't include that extra measurement and simply use the closest interval on the tool. So if we had a ruler with increments of quarter-inches and the true value of length was a quarter and one eighth of an inch, we would only measure a sixteenth of an inch due to the intervals not being precise enough. A measuring instrument with smaller increments would be more precise as there is less uncertainty in the measured value. To maximize precision, minimize increment value.

To get the precision of a linear measurement, find the smallest increment on the tool used to measure the line segment, and halve it. This provides the **absolute error** of measurement. Then, add this value to the measurement and also subtract it. This creates a range of values in which the true length could be.

## Examples of linear measure and precision

Earlier we listed some real-life applications of linear measure and precision. Now, let's take a look at some example questions that you may encounter regarding linear measure and precision.

**Find the length of the line segment AB.**

Working: The zero mark on the ruler is placed on the endpoint A. The large increments go up in 1s, and there are 10 smaller increments between, so each small increment represents 0.1 inches. Point B goes past the 4 mark, and there are 6 smaller increments between 4 and point B, so this means that the length of AB is 4.6 inches.

**A ruler which is divided into half inches measures a line segment of length 7 inches. Find the precision of this measurement.**

Working: The ruler has increments of half inches, so we halve this value to get $\frac{1}{2}\xf72=\frac{1}{4}$. This is the absolute error of measurement. Then, we add and subtract this value to 7 to get a range of values for which the true length could lie within. The ruler is precise within half of an inch, and the measurement could be $6\frac{3}{4}$to $7\frac{1}{4}$.

## Linear measure and precision in geometry

In geometry, points are **collinear** if they lie on the same line. A point C is between points A and B if A, B and C are collinear and the lengths of AC + CB = the length of AB.

Point C is collinear with A and B. Point D is not collinear with A and B. Therefore, point C is between points A and B, while point D is not.

If a point lies between two points, we can use the fact that the linear measure between the end points and the point between them add up to get the length of the whole line segment.

Find the value of x.

Working: We can see that point C is collinear with points A and B. This means that AC + CB = AB. The diagram tells us that the length of AC is 5 inches and the length of AB is 7 inches. Therefore, we subtract 5 from 7 to get the remaining length, which gives us our value for x. 7 - 5 = 2, so x = 2.

Find the value of x.

Working: This example is a little trickier, as we have 3 lengths involving unknowns. However, we can see that point R is between points P and Q, so we can use our knowledge of linear measure in geometry to form an equation. As point R is between points P and Q: PR + RQ = PQ. So, we substitute the values on the diagram into this equation, giving us x + 2 + 3x + 4 = 7x, which can be simplified to 4x + 6 = 7x. We then solve the equation for x by subtracting 4x from both sides, giving us 6 = 3x, and then dividing both sides by two, giving us our answer, x = 2.

## Types of linear measure and precision

When taking linear measurements, there are a range of different measurement tools that we can use. This is because each instrument can have a different precision as their increment values are different. We use the appropriate measuring instrument based on the general size of the line segment we want to measure.

For example, if we wanted to measure the length of a corridor, we would most likely use a ruler with feet or meter increments. This is because the corridor is too big to use smaller measurements and it would be more efficient. For smaller objects, such as an apple, we would use a ruler with inch or centimeter increments as more precision is required to measure smaller lengths correctly and accurately. There are many different tools other than rulers which we can use to measure shorter or longer lengths. Here are some examples:

- Vernier calipers are tools which can be used to measure very small lengths and typically provide a precision of around a thousandth of an inch.
- Micrometers are used for even smaller lengths and have a higher precision than Vernier calipers, generally around a few ten thousandths of an inch.
- Measuring wheels can be used to measure long distances, and have a very low precision as they only notify the user when it has travelled a certain amount of distance, generally a meter or a foot. They work by having a circumference equal to the distance measured, and when the wheel rotates one full rotation, it has travelled one increment of that measurement.
- Laser measurement devices work by emitting a laser from one endpoint of the line segment to the other end. The device calculates the distance based on the time taken for the laser to travel to the endpoint and reflect back.
- Electromagnetic measurement devices use electromagnetic waves to determine the distance between two points.

## Units of linear measure and precision

When taking linear measurements we use units to define certain amounts of lengths. There are two systems for linear measurement: imperial and metric.

Imperial units of linear measure include:

- Inches
- Feet, which are the measurement of 12 inches
- Yards, which are the measurement of 3 feet
- Miles, which are the measurement of 1,760 yards

Metric units work in powers of ten. Metric units of linear measure include:

- Millimeters
- Centimeters, which are the measurement of 10 millimeters
- Meters, which are the measurement of 100 centimeters or 1,000 millimeters
- Kilometers, which are the measurement of 1,000 meters

There are more metric units which can be obtained by either dividing millimeters by 1,000, or multiplying kilometers by 1,000.

Units of linear measurement can be converted by knowing how many of each unit make up another unit. This forms a **ratio** between the two measurements. For example, as each foot is made up of 12 inches, the ratio between the two units could be represented as $\frac{12inches}{1foot}=12$.

Using these ratios, we can convert the units used to express a measurement. To do this unit conversion, we multiply the quantity's original unit by the ratio between that unit and another unit. This is called dimensional analysis. We will take a look at an example to see how this is done.

**How many inches are in 2 yards?**

**Solution****:**

We know that 3 feet are in each yard, so multiply the number of yards by the ratio between feet and yards.

$2yards\times \frac{3feet}{1yard}=6feet$

Each foot is equal to 12 inches, so we multiply the number of feet by the ratio between inches and feet.

$6feet\times \frac{12inches}{1foot}=72inches$

Thus, there are 72 inches in 2 yards. Another way to think of this problem is that we want to convert this linear measurement (2 yards, in this case) to be expressed using the unit of the inch instead. In other words, the quantity itself remains the same, but the units we express it with can be changed using dimensional analysis.

## Linear measure and precision - Key takeaways

**Lines**join two points and extend to infinity.**Line segments**are parts of lines that join two points together.**Linear measure**is the measurement of the length of a line segment.- To define specific amounts of linear measurement, we use
**units**. - We can use measuring tools to find lengths of line segments.
- The
**precision**of a measurement is based on the tool used. The precision of a tool is given by its smallest increment divided by two. - In geometry, points are
**collinear**if they lie on the same line. A point C is between points A and B if A, B and C are collinear and the lengths of AC + CB = the length of AB. - When taking linear measurements, there are a range of different measurement tools that we can use.
- There are two systems for linear measurement: imperial and metric.

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##### Frequently Asked Questions about Linear Measure and Precision

What is linear measure and precision in maths?

**Linear measure**is the measurement of the length of a line segment.- The
**precision**of a measurement is based on the tool used. The precision of a tool is given by its smallest increment divided by two.

What is an example of linear measure?

An example of linear measure would be measuring the length of a football field.

How to measure linear measurements?

Linear measurements can be measured using a measuring instrument, such as a ruler.

What is the basic unit of linear measurement?

The base unit of linear measurement is a meter.

What are the methods of linear measurement?

The different methods of linear measurement are direct linear measurement (for example using a ruler), electromagnetic methods, and optical methods (using a laser to find length).

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