HCPSS Worthwhile Math Task
Similar Parts Task
Common Core Standard
G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a
triangle divides the other two sides proportionally, and conversely; the Pythagorean Theorem
proved using triangle similarity.
Common Core Traditional Pathway: Geometry, Unit 2
[Note: The unit is only needed for Algebra I, Geometry, or Algebra II]
The Task
Figure 2-1 below shows PQR and four identical copies. Arrange the four copies of PQR so
as to create a triangle similar to it. (You may copy, cut out, and use the copies.) Try the same
method for XYZ in Figure 2-2 when finished.
Figure 2-1
Figure 2-2
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
Facilitator Notes
1. This task can be used to discover or prove many geometric theorems relating to similar
triangles. In particular, it illustrates that a line parallel to one side of a triangle divides
the other two sides proportionally, and conversely.
2. Students should be encouraged to discuss their methods for making a larger, similar
triangle out of the 4 smaller ones. The teacher should ask questions that cause them to
consider the similar triangle properties involved.
3. After reasoning about properties of similar triangles by putting pieces together (like a
puzzle), this problem asks students to reverse the process and to undo the dissection by
considering a “pieced-together’ figure first. This gives students the opportunity to work
backwards from a final state—balancing exploration and reflection.
Follow-Up Questions
1. Did the same method work for both triangles? If not, explain why not. If so, will this
method work for any triangle? Explain your answer.
2. Now cut out an arbitrary triangle from a piece of patty paper. Consider this triangle to be
the “final product”—that is, a large triangle that has been formed out of 4 smaller, similar
triangles. Fold the paper so as to reveal the 4 smaller triangles. Has your answer to the
last question changed based on this exploration? Why or why not?
Solutions
Figure 2-1 Sample Solution:
Figure 2-2 Sample Solution:
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
Follow Up Questions:
1. State your findings about similar triangles in your own words.
2. How has this investigation helped you to understand the relationship between
corresponding parts of similar triangles? Write down any new or reaffirmed
understandings that you gained.”
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.