A deceptively simple question setup leads to a number of attack methods and a surprisingly sophisticated solution set in this open-ended problem. Young geometers of different strengths can go about defining the solutions graphically, algebraically, or even through abstract algebra. The basics of triangle geometry are all that is really required as far as prerequisite knowledge, so this is a rare early-class open-ended proof to really challenge your learners.

Concepts

distance, proofs, triangles, area, solutions, solution set

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Additional Tags

math

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Instructional Ideas

• Direct groups of students to use different methods and then have them present their findings and method to the class
• Have students create similar generalized questions for one another; fix height and area of a rectangle, or apothem and area of some regular polygon

Classroom Considerations

• Some solution methods (matrices, for example) lend themselves to more calculator work; learners might need encouragement to work relevant pieces out on paper

Pros

• Simple prerequisites but deep thinking required, leveling much of the playing field for a class with a variety of entry skill levels
• Room for different methods and interpretations to the same answer models good mathematical thought and rigor

Cons

• Student worksheet not separated from teaching notes and answer key

Common Core

HSG-GPE.B.5