Learning Math: Measurement
Indirect Measurement and Trigonometry Part A: Indirect Measurement With a Transit (40 minutes)
Session 5, Part A
In This Part
- Similar Triangles
- Using Similar Triangles
- Measuring Distances
One way to measure indirectly is to use similar triangles. In similar figures, corresponding angles are congruent, and corresponding sides (or segments) are in proportion. In similar triangles, however, one or the other will suffice. In other words, if the corresponding sides alone are in proportion, the triangles must be similar.
These triangles have proportional corresponding sides (a ratio of 1:2):
Likewise, if the corresponding angles alone are congruent, the triangles must be similar. Notice that because the sum of angles in a triangle must be 180 degrees, we only need to know that two of the corresponding angles are congruent to know that they are similar.
These triangles have congruent corresponding angles:
Using Similar Triangles
If we have actual measures for at least one of the similar triangles, and we also know the scale factor or ratio that links the two triangles, we can use proportional reasoning to determine the measure of the unknown side(s) on the other triangle. This unknown side corresponds to the object or feature of the object we’re trying to measure.
In the following example, triangle ABC is similar to triangle DEF. In the following example, triangle ABC is similar to triangle DEF. To find the length of x, you set up a proportion as shown below. (Notice that x is perpendicular to side a, which will help us with calculations.)
When we use similar triangles to measure indirectly, we usually collect some measurements from a triangle that can be imagined using landmarks (the lengths of some of the sides and/or the measure of some of the angles) and then draw a similar triangle on paper. We need adequate information to make sure that we are dealing with a unique triangle; knowing the length of one side of a triangle would not be enough information to draw a similar triangle, but knowing two angles and one side would be. We also need to know the ratio that links the corresponding sides in the two similar triangles as you’ve seen above. We will explore this process in greater detail in the next section
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:
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 Δ ABC (we’ll call it Δ 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 drawΔ 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:
AB (10 m) = BC(x)
A1B1 (5 cm) B1C1 (8 cm)
The length of BC is 16 m.
- Use the technique of similar triangles to determine the distance between two objects outside.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.
- Decide on a scale factor (similarity ratio) and draw a similar triangle. (You will need to use a protractor.)
- Find the approximate distance to your object from one of your sight points.
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?
You can find this segment on the session video approximately 5 minutes and 53 seconds after the Annenberg Media logo.
Why do you think we use similar triangles rather than other similar figures for this indirect measurement?
Think about how many measurements are needed in order to have a unique triangle.
In Problem A1, what other distances could you find indirectly using your similar triangle?
Explain in your own words the relationship between similarity and indirect measurement.
It may not be clear that an infinite number of similar triangles can be drawn to correspond to the original triangle. This is because a ratio can be established between corresponding sides, no matter what the size of each of the triangles (as long as the angles in both triangles are congruent). If you are working in a group, discuss this idea and make sure that everyone in the group understands how to set up a proportion in which one ratio is equal to another ratio.
Answers will vary. Here is one example:
- The similarity ratio here is 5 m:1 cm. Here is the similar triangle:
- The distance to the lighthouse (the length BC) is 53.5 m
The three measurements (angle, side, angle) determine a unique triangle; proving that two triangles are similar requires only two more measurements (the two angles in the second triangle). Also, as we’ve seen in Session 4, every polygon can be divided into triangles, which can be regarded as its basic building blocks. Therefore, triangles will work in every situation, which is why we use them instead of any other polygon.
Using the same triangle, you could find the distance from your other sight point to the tree.
Answers will vary, but here is one example: An indirect measurement can be taken when two figures are known to be similar and when a known measurement is taken from each figure. The ratio of this measurement establishes a scale factor for any other measurements that compare the two similar figures.
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.