Median altitude geometry1/5/2024 ![]() We are wanting to show that the green median and the red median are congruent. Hopefully, after some investigation in GSP, we suspect that the two medians drawn to the two congruent sides of the isosceles triangle are in fact congruent. Need to explore medians of an isosceles triangle in GSP? Click here. ![]() We are hoping that two of our medians will be congruent and thus the medians of an isosceles triangle will form an isosceles triangle. Substituting into two of our three formulas we have:Īnd then solving each for the median we generate the three following formulas:Īnd now, it should be quite obvious that having generated the same formula for each median that the medians of an equilateral triangle are in fact congruent and therefore the triangle formed by these congruent medians is equilateral.Īnd so, we are now definitively convinced that the triangle formed by the medians of an equilateral triangle is itself equilateral.ĭoes the same hold true for an isosceles triangles? Well, I am glad you asked.Ĭonsider an isosceles triangle and construct its medians. We know that all sides of an equilateral triangle are congruent and so we can call each side a. Calculating the area using each of the three altitudes we have:īM B is an altitude with respect to side b of our triangle.ĬM C is an altitude with respect to side c of our triangle.ĪM A is an altitude with respect to side a of our triangle. We know that the area of our triangle is constant no matter which altitude we use to calculate. Īnd so segment AM A is the perpendicular drawn from a vertex to the line containing the opposite side of the triangle, which in fact means that AM A is an altitude with respect to side a of our triangle.īy similar argument we are able to show that each median is altitude of the triangle. If two angles are supplementary and congruent, then they must be right angles.īecause segments AM A and BC form right angles at M A, we can conclude that they are perpendicular at M A. Thus, by the Side-Side-Side triangle congruence postulate.īecause corresponding parts of congruent triangles are congruent.įrom the angle addition postulate, we know that, which in turn tells us that and are supplementary angles. ![]() Take any median, in this case we’ve chosen m a (whose endpoints are A and M a ).īecause the sides of an equilateral triangle are congruent.īecause the midpoint of a segment divides that segment into two congruent segments. In an equilateral triangle the medians of the triangle are also the altitudes. It should be easy to see that all three medians are congruent.Ĭlick here to explore the relationship between the three medians of an equilateral triangle in GSP. Department ofMathematics and Science EducationĬharacterization of Medial Triangles formed from Equilateral and Isosceles TrianglesĬharacterization of Triangles whose Medians form Right TrianglesĬonsider an equilateral triangle and construct its medians.
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