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Students will use prior knowledge of similar triangles to:
➢ Investigate the relationship between a triangle’s midsegment and its base.
➢ Develop the rationale for why particular relationships exist in a triangle.
➢ Develop logical reasoning in a geometric context.
This two page exploration allows students to discover the relationship between a triangle's midsegment and it's base, while reviewing similar triangles, congruent triangles, and properties of parallel lines cut by a transversal. Students must answer open ended questions that truly require them to think about why certain relationships exist.
Part I - Students are given one requirement for their triangle: BC is 16 units. Each group randomly plots point A above BC so that each group has a different triangle (this leads to the discussion of why every group's midsegment is equal). Students then construct the midsegments of AC and AB, connect them, and find its length. Student should discover that the midsegment created is 8 units to matter where they placed A. At the end of part I, they must predict the ratio of the area of the smaller triangle ADE to the larger triangle ABC. Most students will say 1:2.
Part II - Students find the midpoints of the 2 other sides to create the other 2 midsegments. They find the length of the midsegments and must predict the length of each midsegment's base.
Part III - Students discover the true relationship of the area of triangle ADE to the area of triangle ABC, and then discuss why the relationship is 1:4 instead of 1:2.
The exit ticket is one problem in which students must use the pythagorean theorem to find the hypotenuse of the right triangle and the length of it's midsegment.