Math 107Z
Remote Measurement Project
Summary of information on similar triangles
As yet another reference for the relationship between triangles which are similar, consider the triangles BCD and B’C’D’ which I
constructed in Geometer’s SketchPad such that the measures of angles B, C, and D are equal to the measures of angles B’, C’, and
D’, respectively.
C
C’
B
D
B’
D’
These triangles are defined to be similar (having the same shape, but not necessarily the same size). If these two triangles were
congruent, the lengths of their corresponding sides would be equal, and their areas would be equal. We get something almost as
good with similar triangles; the ratios of the lengths of their corresponding sides is a constant (called the scale factor), and the ratio
of their areas is the square of this constant. In the triangles shown above, this means that
BC
CD
DB



B' C' C' D' D' B'
Area(BCD)
Area(B' C' D' )
Here are a few mathematical short-cuts for determining when two triangles are similar:
• (AA) If two angles of one triangle are congruent to two corresponding angles of another triangle, then the two triangles are
similar.
• (SSS) If all corresponding sides of two triangles are in proportion, then the triangles are similar.
• (SAS) If two pairs of corresponding sides are in proportion and the angles included between them are congruent, then the
triangles are similar.
Project Task
You and your group are to measure the height of ANY tree on Campus that appears taller than a
4-story building. I suggest that you consider a tree like the redwood tree that is due East of the
Main library entrance, just west of the music building. Why? This tree was trimmed two years
ago, and the top is easy to see, and there is a fair amount of open space on one side of this tree
for you to conduct your measurements.
GATHER DATA (3/10/99)
How tall is the tree in meters? Use the two different measurement techniques
that were suggested in class and perform them each from two different
observation points. (So you will have 4 sets of measurement data). Note that
in one of the techniques you must pay special attention to the general context of
your measurement – e.g., is the ground level between the observation point and
the tree? So in addition to numerical data, create a scaled drawing that shows
the geometry of the situation.
ANALYSIS AND CONCLUSION
Based on your data, determine a reasonable interval for the actual height of
the tree. For example you might claim something like, “the tree we measured is
between 15 and 17 meters tall.” You must be able to justify this range BASED
ON YOUR DATA. If you decide to throw out some of your measurements,
make sure you explain why.
INDIVIDUAL REPORT (9 points) Due Wednesday, March 24.
Each group member must write a summary of your data gathering and analysis
stages. The report should contain three sections:
Description of the task
What tree was measured (include the location on campus)?
How was the tree height measured?
Report of the data
Show 4 drawings with measurements and height calculations.
Analysis of the data
Write a paragraph in which you state your group decision about a
reasonable interval for the actual height of the tree and your justification
for this decision.
GROUP REPORT (3 points)
On Monday, 3/22/99 Your group will have 20 minutes of preparation time to get
a 5-minute group presentation ready. I will have overhead sheets and pens for
your data as well as large flip-chart paper for your scale drawing.
GROUP QUESTIONS (3 points)
On Monday, 3/22/99 we will follow-up the presentations with a discussion of
error propagation in measurement. Each group will be asked to analyze error
propagation on one of their measurements. Each group will also be sent out to
measure another group’s tree with a third technique as a check of that groups’
work.