ANCIENT RUINS
Enduring Understanding: Develop a better understanding of how similarity can be used to
determine lengths of the sides of a triangle.
Essential Questions:
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How does a person determine that two triangles are similar?
What does similarity mean?
How are ratio and proportion used to solve problems that deal with similarity?
What is needed to determine the perimeter of a figure?
How can the Pythagorean Theorem assist in finding the missing lengths of the sides of a right
triangle?
How can the ratios of the areas and volumes be determined when the scale factor of two
similar figures is known?
What are the properties of an altitude of a triangle?
What happens to the area and volume of a three-dimensional solid when the length, width or
height is changed?
Lesson Overview:
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Before allowing the students the opportunity to start the activity: access their prior
knowledge with regards to properties of triangles, parallel lines and scale factor. Warm-up
exercises, discussion in collaborative groups, problems on the wall around the room can be
used.
Have the students read through the initial problem and lead a class discussion to clarify the
vocabulary (i.e. excavate).
Discuss proportionality, especially as it applies to question #5 of the scenario.
What are the properties of a triangle?
What is being asked by the questions in the problem? How do you decode what the problem
is asking you to do?
How can the students make their thinking visible?
Use resources from your building. Technology could be used to support this investigation—
especially sketchpad or cabri geometry.
EALRs/GLEs:
1.1.4
1.2.5
1.3.1
1.3.2
2.2.1
2.2.2
2.2.3
5.1.1
Item Specifications: NS02; ME03; GS01; SR02; MC01
Assessment:
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Use WASL format items that link to what is being covered by the classroom activity
Include multiple choice questions
Ancient Ruins
Adapted from the Charles A. Dana Center at the University of Texas at Austin, © 2002
Archaeologists flying over a remote area in the interior of Mexico saw what appeared to be the ruins
of an ancient ceremonial temple complex below. This diagram shows what they saw:
Preparation Room
Main
Key:
- - - - - - crumpled
______ existing
Temple
Based on their photographs they believed that some of the walls were still intact. These are drawn as
solid segments, and the walls that had crumpled or had obviously been there are drawn as dashed
segments. Before they can excavate the site they need to construct an accurate scale drawing or
model. Based on their altitude flying over the ruins and the measurements made on the photograph,
they generated the following drawing of the ruins:
The lengths, in feet, of the walls AB , AC , DE , and DF were 19.5, 22.5, 78 and 72, respectively.
AB / / DE and DF

CE .
To plan for the excavation they need to know a number of things about the site.
1. The archaeologists estimate it will require 45 minutes to an hour to excavate each foot of the
exterior walls of the temple site. Approximately how long will it take to complete this task, if
they have to expose all of the walls? Show all work you used to determine this.
2. The entrance into the ceremony preparation room appears to be along wall BC and directly
opposite vertex A.
a. What would this mean geometrically? _________________________________________
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b. What is the distance from point A to the entrance? _______________________________
Show how you determined this answer.
c. Were any of the exterior walls of this room perpendicular to each other? ______________
d. How do you know whether or not the walls were perpendicular?
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3. The archaeologists believe that about 9 square feet of space was needed in the main temple
for each person.
a. How many people could occupy the temple (triangle CDE)? Show all of your work.
b. How many priests and assistants could occupy what appears to be the ceremony
preparation room (triangle ABC)? Show all of your work.
4. In the proposal that the archaeologists must write to excavate the site, they must compare the
perimeters and areas of the ceremony preparation room to those of the main temple. Use
words, numbers and/or diagrams to support how this could be done.
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5. It is believed that each of the two rooms was built in the shape of a triangular pyramid.
a. What can be said about the capacity of the rooms?
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b. Is it possible to determine their capacity? ___________________
c. Why do you know this?
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6.
Which represents the relationship of the angles of PQR from smallest to largest?
O A.
O B.
O C.
O D.
R, Q, P
R, P, Q
Q, R, P
P, R, Q
7. The club members hiked 13 kilometers north and 14 kilometers east, but then went directly home
as shown by the dotted line.
Which is the distance they traveled to get home?
OA.
5.2 km
O B.
15.0 km
O C.
19.1 km
O D.
27.0 km