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

No Core Resource for this Big Idea.

Consider using the Instructional Routines linked below for teaching towards this Big Idea.

    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.

      All Resources From: