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Distance Between Two Points

Positive length between points.

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Distance Between Two Points

The average adult human body can be measured in “heads.” For example, the average human is 7-8 heads tall. When doing this, keep in mind that each person uses their own head to measure their own body. Other interesting measurements are in the picture below.

What if you wanted to determine other measurements like the length from the wrist to the elbow or the length from the top of the neck to the hip? After completing this Concept, you will know how to accurately measure using many different units of measurement.

Watch This

CK-12 Foundation: Chapter1DistanceBetweenTwoPointsA

James Sousa: Ruler Postulate and Segment Addition Postulate

Guidance

Distance is the length between two points. To measure is to determine how far apart two geometric objects are. Inch-rulers are usually divided up by -in. (or 0.125 in) segments. Centimeter rulers are divided up by -centimenter (or 0.1 cm) segments.

The two rulers above are NOT DRAWN TO SCALE. Anytime you see this statement, it means that the measured length is not actually the distance apart that it is labeled. You should never assume that objects are drawn to scale. Always rely on the measurements or markings given in a diagram.

The Ruler Postulate states that the distance between two points will be the absolute value of the difference between the numbers shown on the ruler. The ruler postulate implies that you do not need to start measuring at “0”, as long as you subtract the first number from the second. “Absolute value” is used because distance is always positive.

Before we introduce the next postulate, we need to address what the word “between” means in geometry.

is between and in this picture. As long as is anywhere on the segment, it can be considered to be between the endpoints.

The Segment Addition Postulate states that if , , and are collinear and is between and , then .

The picture above illustrates the Segment Addition Postulate. If and , then must equal or 17 cm. You may also think of this as the “sum of the partial lengths, will be equal to the whole length.”

In Algebra, you worked with graphing lines and plotting points in the plane. At this point, you can find the distances between points plotted in the plane if the lines are horizontal or vertical. If the line is vertical, find the change in the coordinates. If the line is horizontal, find the change in the coordinates.

Example A

What is the distance marked on the ruler below? The ruler is in centimeters.

Find the absolute value of difference between the numbers shown. The line segment spans from 3 cm to 8 cm.

The line segment is 5 cm long. Notice that you also could have done

Example B

Make a sketch of , where is between and .

Draw first, then place somewhere along the segment.

Example C

What is the distance between the two points shown below?

Because this line is vertical, look at the change in the coordinates.

The distance between the two points is 6 units.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter1DistanceBetweenTwoPointsB

Vocabulary

Distance is the length between two points. To measure is to determine how far apart two geometric objects are.

Guided Practice

1. Draw , such that .

2. If and , what is ?

3. Make a sketch that matches the description: is between and . is between and . and . Then, find and .

4. For , suppose that is between and . If and , find the lengths of and .

5. What is the distance between the two points shown below?

Answers:

1.To draw a line segment, start at “0” and draw a segment to 3.825 in. Put points at each end and label.

2. Use the Segment Additional Postulate. , so , or . So, .

3. Interpret the first sentence first: is between and .

Then add in what we know about : It is between and .

To find , we know it is equal to , so .

4. Use the Segment Addition Postulate and then substitute what we know.

5. Because this line is horizontal, look at the change in the coordinates.

The distance between the two points is 7 units.

Interactive Practice

Practice

For 1-4, use the ruler in each picture to determine the length of the line segment.

  1. Make a sketch of , with between and .
  2. If is in the middle of , where exactly is it located? If , what is and ?
  3. For three collinear points, between and .
    1. Draw a sketch.
    2. Write the Segment Addition Postulate.
    3. If and , what is ?
  4. For three collinear points, between and .
    1. Draw a sketch.
    2. Write the Segment Addition Postulate.
    3. If and , what is ?
  5. Make a sketch that matches the description: is between and . is between and . and . Find and .

For 10-14, Suppose is between and . Use the Segment Addition Postulate to solve for . Then find the length of each segment.

  1. Draw four points, and such that (HINT: and should NOT be collinear)

For 16-19, determine the vertical or horizontal distance between the two points.

Vocabulary

Absolute Value

Absolute Value

The absolute value of a number is the distance the number is from zero. Absolute values are never negative.
measure

measure

To measure distance is to determine how far apart two geometric objects are by using a number line or ruler.

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