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62. Use a vector argument to prove that the segment connecting the midpoints of two sides of a triangle is parallel to the third side of the triangle and half of its length.

Short Answer

Expert verified
  • A triangle midsegment is a segment that connects the midpoints of two triangle sides.
  • Any two sides of a triangle's midpoints are parallel to the third side.

Step by step solution

01

Introduction

Consider the triangleABCwith verticesa,band c

The objective is to use vector argument to prove that segment connecting mid-points of two sides of a triangle is parallel to the third side of the triangle and half of its length.

Consider the triangle ABCwith vertices representing the vectors a,b, and cas shown below.

Triangle ABC


02

Given Information.

The mid-point between the vectors aand bis:

a+b2

The mid-point between the vectors aand cis:

a+c2

Now, compare the line segment from Bto Cand from Dto E.

The vector from Bto Cis:

BC=c-b

03

Explanation (part a)

The vector from DtoEis:

DE=a+c2-a+b2

=a+c-a-b2=c-b2

04

Explanation (part b)

The vector DEis half of the vector BCand is in the direction of the vector BC. Therefore, the vector DE is parallel to the vectorBC

Hence, segment connecting mid-points of two sides of a triangle is parallel to the third side of the triangle and half of its length.

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