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(a) The line segment connecting the midpoints of any two sides is parallel to the third side, and its length is exactly half the length of the third side. (b) The three medians intersect in a single point, and this point divides each median in a 2:1 ratio. (Recall that a median of a triangle is a line segment connecting one corner to the midpoint of the opposite side.)

(a) The line segment connecting the midpoints of any two sides is parallel to the third side, and its length is exactly half the length of the third side. (b) The three medians intersect in a single point, and this point divides each median in a 2:1 ratio. (Recall that a median of a triangle is a line segment connecting one corner to the midpoint of the opposite side.)

Elementary Geometry for College Students
Elementary Geometry for College Students
6th Edition
ISBN:9781285195698
Author:Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:Daniel C. Alexander, Geralyn M. Koeberlein
Chapter10: Analytic Geometry
Section10.CT: Test
Problem 22CT
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Use vector-based methods to prove the following facts concerning a triangle in the plane. (a) The line segment connecting the midpoints of any two sides is parallel to the third side, and its length is exactly half the length of the third side. (b) The three medians intersect in a single point, and this point divides each median in a 2:1 ratio. (Recall that a median of a triangle is a line segment connecting one corner to the midpoint of the opposite side.) (Hint: Both statements are invariant under translations, rotations, and scalings of the triangle, so you may argue that it suffices to consider a triangle with vertices (0,0), (1,0), and (a, b) for some real numbers a and b to simplify your work.)
Transcribed Image Text:Use vector-based methods to prove the following facts concerning a triangle in the plane. (a) The line segment connecting the midpoints of any two sides is parallel to the third side, and its length is exactly half the length of the third side. (b) The three medians intersect in a single point, and this point divides each median in a 2:1 ratio. (Recall that a median of a triangle is a line segment connecting one corner to the midpoint of the opposite side.) (Hint: Both statements are invariant under translations, rotations, and scalings of the triangle, so you may argue that it suffices to consider a triangle with vertices (0,0), (1,0), and (a, b) for some real numbers a and b to simplify your work.)
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