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Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

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BuyFindarrow_forward

Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

In Exercises 1 to 17, complete an analytic proof for each theorem.

The median of a trapezoid is parallel to the bases of the trapezoid and has a length equal to one-half the sum of the lengths of the two bases.

To determine

To answer:

The analytic proof for the given theorem “The median of a trapezoid is parallel to the bases of the trapezoid and has a length equal to one-half the sum of the lengths of the two bases”

Explanation

Given theorem is,

The median of a trapezoid is parallel to the bases of the trapezoid and has a length equal to one-half the sum of the lengths of the two bases.

The above graph shows the trapezoid ABCD.

AB is the lower base and DC is the upper base.

P is the midpoint of AD and Q is the midpoint of BC.

Joining the points P and Q as shown in the above figure.

Coordinates of the trapezoid is A(0, 0), B(a+b+c, 0), C(a+b,d) and D(a,d).

Length of the upper base DC is a below,

DC = (a-a+b)2+(d-d)2

DC = (a-a-b)2

DC = b2=b

Length of the lower length AB is as below,

AB = a+b+c-0)2+(0-0)2

AB = (a+b+c)2=a+b+c

Midpoint of AD is as below,

AD=P=a+02,d+02

AD=P=a2,d2

Similarly, the midpoint of BC is as below,

BC=Q=a+b+c+(a+b)2,d+02

BC=Q=2

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