Big Idea:

Big Idea 3

Medians, altitudes, or perpendicular bisectors intersect at a point of concurrency uniquely positioned in relation to the triangle.

1 week

Evidence of Understanding

  • construct medians and explain why they intersect at the centroid
    • make conjectures about the relationship of the segment lengths that result from constructing 3 medians and prove that the segment lengths are in a ratio of 2:1
    • make conjectures about the relationship of the areas of each triangle formed and prove they are the same (connection to center of mass)
    • find missing values in a triangle that involve the median
    • use tools and counterexamples to prove statements about triangles
       
  • apply perpendicular bisectors to identify the circumcenter
    • construct perpendicular bisectors and apply transformations to justify the location of the circumcenter
    • make conjectures and justify why if two perpendicular bisectors intersect each other, then the third perpendicular bisector would intersect at the same point
      • show the point of intersection is equidistant to the vertices of each side of the triangle it is perpendicularly bisecting
         
  • construct altitudes and use them to determine the orthocenter
    • prove the triangles formed by constructing the altitude are similar
    • analyze the relationship between the orthocenter of a medial triangle and the circumcenter of its larger triangle and prove why they are the same point
      • explain how the perpendicular bisectors of the larger triangle are the same as the altitudes of the medial triangle
    • make conjectures, using tools, about the location of the orthocenter in relationship to the triangle (right, acute, obtuse) and how the angle measures impact the location
      • explain why the orthocenter of a right triangle always lies at the vertex of the triangle’s right angle

Develop conceptual understanding:

median, centroid, perpendicular bisectors, circumcenter, altitude, orthocenter

Supporting terms to communicate:

vertex, midpoint, opposite side, proportional, ratio, scale factor, area, similar, equidistant, right, obtuse, acute, reflection, rotation, dilation, center of dilation, corresponding
Core Resource
A core resource supports multiple days of instruction. COMING SOON!
    Instructional Routine: Connecting Representations

    These tasks are embedded within the instructional routine called Connecting Representations.

    • Centroid, Circumcenter, Orthocenter
      Visualize what constructions of the centroid, circumcenter, and orthocenter would look like based on descriptions of those constructions.
      Resource:
      Centroid, Circumcenter, Orthocenter

      Visualize what constructions of the centroid, circumcenter, and orthocenter would look like based on descriptions of those constructions.

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