Hospital Locator using Special Points in a Triangle

This is an activity designed to help students see a real-world application for finding the special points of a triangle. There is a need locating a hospital that could be used by 3 cities which requires the students to find the circumcenter and incenter and determine which would be better. This could be used to help students see the relevance of circumcenters and incenters. Several guiding questions are given, but there may need to be additional development depending on the teacher's needs.

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Hospital Locator using Special Points in a Triangle

Standards & Objectives

Learning objectives: 

Students will:

  • Construct the circumcenter of a triangle.
  • Construct the incenter of a triangle.
  • Explain how the circumcenter and incenter involve bisecting the triangle’s sides and angles, respectively, and why such a bisection process makes sense.
Essential and guiding questions: 
  • Why does it make sense that the circumcenter would lie on the perpendicular bisectors of the sides?
  • Why does it make sense that the incenter would lie on the angle bisectors?

Lesson Variations

Blooms taxonomy level: 
Understanding
Extension suggestions: 
  • Is it possible for the circumcenter and the incenter of a triangle to be the same point? Explain. [The circumcenter and the incenter would be the same point when all of the perpendicular bisectors of the sides and the angle bisectors of the triangle coincide. This happens when the triangle is an equilateral triangle. So, yes, it is possible.]
  • Perpendicular bisectors of the sides led to the circumcenter. The bisecting of the angles led to the incenter. What happens if we simply construct a line through the midpoint of each side and the opposite vertex? [The three lines intersect in a unique point. Subsequent discussion may lead to the description of the lines as the medians and the intersection point as the centroid. The sense of "half" comes into the discussion because each median divides the area of the original triangle in half, as can be seen in the Incenter-Incircle Tool. (Because M is the midpoint of the base, it follows that the area of triangles AMB and CMB will always be equal. Both have height BM, and the bases of the triangles are equal (AM = CM), so the areas are equal: ½(AM)(MB) = ½(CM)(MB).]
     

Helpful Hints

  • Computers or tablets with internet connection (1 for class OR 1 for each pair or small group of students)
  • Hospital Problem Overhead (1 copy)
  • Cities Triangle Overhead (at least 1 copy for each student)
  • Hospital Map Activity Sheet (at least 1 copy for each student)
  • Blank transparencies (several per pair or group of students)
  • Compass, straight edge, ruler, protractor (1 set per group)

References

Contributors: 
Eric Istre
Originator: 
Ayers Institute for Learning & Innovation