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Learning Math Home
Measurement Session 5:  Measurement and Trigonometry
 
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Session 5, Part A:
Indirect Measurement With a Transit

In This Part: Similar Triangles | Using Similar Triangles | Measuring Distances

When measuring outdoors, it's relatively easy to measure the size of different angles. For example, if we want to make a scale drawing of a particular location, we can work with the angles formed by imaginary lines joining trees, buildings, and other landmarks. To take such horizontal and vertical angle measurements, civil engineers use an instrument called a transit. We will use a homemade transit to measure horizontal angles.

Suppose we want to find the distance across a field to a tree. We'll make the base of the imaginary right triangle the side of the field where we're standing, and the tree the opposite vertex of the triangle. We can imagine an infinite number of triangles. Here is one possibility: Note 3

Even when you measure indirectly, you still have to take some direct measurements. First you must establish the measure of at least two of the angles in a triangle. (Why don't you have to measure the third angle?) Then you must physically measure one side of the triangle so that you can establish a proportional relationship between the side you've measured and the corresponding side in the similar triangle.

To take such measurements, you can use a homemade transit for the angles and a trundle wheel for the distance between them. You can make a transit with a straw, a metric ruler, a protractor, a pushpin, and some tape.

To use the transit, stand at each endpoint of the base of your imaginary triangle and hold the transit at eye level. Move the straw to line up with the object under scrutiny, and read the angle measure off the protractor.

To try this yourself, go outside and find a tree across a field or parking lot. Set up an imaginary line along the side of the open space opposite the tree by putting markers down for points A and B. From point B, the tree should appear to be directly in front of you. Using the transit, sight the tree (which will be point C) across the field. You want point B to be perpendicular to both points A and C (namely, B should be a 90-degree angle). You may have to move point B a bit on your base line to make sure that you have a right angle. Next, use the trundle wheel to find the distance between points A and B. Make sure that the distance is at least 10 m (you may have to move point A). Finally, stand at point A and sight the tree (point C) in the distance, using the transit. Draw a sketch of ABC and record the angle measures for A and B and the actual distance between points A and B. Notice that we don't know the distance between points B and C and between points A and C at this time.

Now your triangle might look something like this:

Next, you need to draw a triangle similar to ABC (we'll call it A'B'C'). The length of AB determines the similarity ratio or scale factor, so you want to pick a convenient scale; for example, 1 cm on the drawing could equal 2 m in the real world. Use your scale factor to determine the length of A'B' and to draw A'B'C'. Next, measure B'C' on your scale drawing and set up a proportion to find the corresponding measurement (BC) in the original triangle. Here is an example using the scale 1 cm:2 m:

The length of BC is 16 m.

Problem A1

Solution  

Use the technique of similar triangles to determine the distance between two objects outside.

a. 

Use your transit to help you construct a right triangle and measure angles. Remember to measure the distance between your two sight points. Sketch a triangle that represents the angle and side measures.

b. 

Decide on a scale factor (similarity ratio) and draw a similar triangle. (You will need to use a protractor.)

c. 

Find the approximate distance to your object from one of your sight points.



video thumbnail
 

Video Segment
In this segment, Susan and Jonathan indirectly measure the distance across a parking lot to a nearby tree. They use a trundle wheel and a transit to measure one side and one angle of a triangle, and then set up a similar triangle to calculate the unknown distance.

What are some advantages and disadvantages of this type of indirect measuring?

If you are using a VCR, you can find this segment on the session video approximately 5 minutes and 53 seconds after the Annenberg Media logo.

 

 

Problem A2

Solution  

Why do you think we use similar triangles rather than other similar figures for this indirect measurement?


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Think about how many measurements are needed in order to have a unique triangle.   Close Tip

 

Problem A3

Solution  

In Problem A1, what other distances could you find indirectly using your similar triangle?


 

Problem A4

Solution  

Explain in your own words the relationship between similarity and indirect measurement.


Next > Part B: Measuring Heights of Tall Objects

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