Chapter 10.4, Problem 33E

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7th Edition

Alexander + 2 others

ISBN: 9781337614085

Textbook Problem

Would the theorem of Exercise 7 remain true for a concave quadrilateral like the one shown?

*Exercise 33.*

To determine

**To answer:**

The analytic proof for the given theorem “Would the theorem of Exercise 7 remain true for the given concave quadrilateral”

Explanation

Given figure is,

The above figure shows the quadrilateral.

And the theorem of Exercise 7 is “The line segments that join the midpoints of the consecutive sides of a quadrilateral form a parallelogram”

To prove “The line segments that join the midpoints of the consecutive sides of a quadrilateral form a parallelogram” for the given figure, follow as below,

The coordinates of the quadrilateral ABCD is as shown in the figure,

Let R, S, T, V are the midpoint of AB, BC, CD and AD.

Joining the R, T, TV and VR as shown in the figure.

Midpoint of AB = S =

Midpoint of AB = S =

Midpoint of AB = S =

Midpoint of BC = T =

Midpoint of BC = T =

Midpoint of BC = T =

Midpoint of CD = V =

Midpoint of CD = V =

Midpoint of CD = V =

Midpoint of AD = R =

Midpoint of AD = R =

Midpoint of AD = R =

Determining the slopes as below,

Sect-10.1 P-1ESect-10.1 P-2ESect-10.1 P-3ESect-10.1 P-4ESect-10.1 P-5ESect-10.1 P-6ESect-10.1 P-7ESect-10.1 P-8ESect-10.1 P-9ESect-10.1 P-10E

Sect-10.1 P-11ESect-10.1 P-12ESect-10.1 P-13ESect-10.1 P-14ESect-10.1 P-15ESect-10.1 P-16ESect-10.1 P-17ESect-10.1 P-18ESect-10.1 P-19ESect-10.1 P-20ESect-10.1 P-21ESect-10.1 P-22ESect-10.1 P-23ESect-10.1 P-24ESect-10.1 P-25ESect-10.1 P-26ESect-10.1 P-27ESect-10.1 P-28ESect-10.1 P-29ESect-10.1 P-30ESect-10.1 P-31ESect-10.1 P-32ESect-10.1 P-33ESect-10.1 P-34ESect-10.1 P-35ESect-10.1 P-36ESect-10.1 P-37ESect-10.1 P-38ESect-10.1 P-39ESect-10.1 P-40ESect-10.1 P-41ESect-10.1 P-42ESect-10.1 P-43ESect-10.1 P-44ESect-10.1 P-45ESect-10.1 P-46ESect-10.1 P-47ESect-10.1 P-48ESect-10.1 P-49ESect-10.1 P-50ESect-10.1 P-51ESect-10.2 P-1ESect-10.2 P-2ESect-10.2 P-3ESect-10.2 P-4ESect-10.2 P-5ESect-10.2 P-6ESect-10.2 P-7ESect-10.2 P-8ESect-10.2 P-9ESect-10.2 P-10ESect-10.2 P-11ESect-10.2 P-12ESect-10.2 P-13ESect-10.2 P-14ESect-10.2 P-15ESect-10.2 P-16ESect-10.2 P-17ESect-10.2 P-18ESect-10.2 P-19ESect-10.2 P-20ESect-10.2 P-21ESect-10.2 P-22ESect-10.2 P-23ESect-10.2 P-24ESect-10.2 P-25ESect-10.2 P-26ESect-10.2 P-27ESect-10.2 P-28ESect-10.2 P-29ESect-10.2 P-30ESect-10.2 P-31ESect-10.2 P-32ESect-10.2 P-33ESect-10.2 P-34ESect-10.2 P-35ESect-10.2 P-36ESect-10.2 P-37ESect-10.2 P-38ESect-10.2 P-39ESect-10.2 P-40ESect-10.2 P-41ESect-10.2 P-42ESect-10.2 P-43ESect-10.2 P-44ESect-10.2 P-45ESect-10.2 P-46ESect-10.2 P-47ESect-10.3 P-1ESect-10.3 P-2ESect-10.3 P-3ESect-10.3 P-4ESect-10.3 P-5ESect-10.3 P-6ESect-10.3 P-7ESect-10.3 P-8ESect-10.3 P-9ESect-10.3 P-10ESect-10.3 P-11ESect-10.3 P-12ESect-10.3 P-13ESect-10.3 P-14ESect-10.3 P-15ESect-10.3 P-16ESect-10.3 P-17ESect-10.3 P-18ESect-10.3 P-19ESect-10.3 P-20ESect-10.3 P-21ESect-10.3 P-22ESect-10.3 P-23ESect-10.3 P-24ESect-10.3 P-25ESect-10.3 P-26ESect-10.3 P-27ESect-10.3 P-28ESect-10.3 P-29ESect-10.3 P-30ESect-10.3 P-31ESect-10.3 P-32ESect-10.3 P-33ESect-10.3 P-34ESect-10.3 P-35ESect-10.3 P-36ESect-10.3 P-37ESect-10.3 P-38ESect-10.3 P-39ESect-10.3 P-40ESect-10.3 P-41ESect-10.4 P-1ESect-10.4 P-2ESect-10.4 P-3ESect-10.4 P-4ESect-10.4 P-5ESect-10.4 P-6ESect-10.4 P-7ESect-10.4 P-8ESect-10.4 P-9ESect-10.4 P-10ESect-10.4 P-11ESect-10.4 P-12ESect-10.4 P-13ESect-10.4 P-14ESect-10.4 P-15ESect-10.4 P-16ESect-10.4 P-17ESect-10.4 P-18ESect-10.4 P-19ESect-10.4 P-20ESect-10.4 P-21ESect-10.4 P-22ESect-10.4 P-23ESect-10.4 P-24ESect-10.4 P-25ESect-10.4 P-26ESect-10.4 P-27ESect-10.4 P-28ESect-10.4 P-29ESect-10.4 P-30ESect-10.4 P-31ESect-10.4 P-32ESect-10.4 P-33ESect-10.4 P-34ESect-10.4 P-35ESect-10.5 P-1ESect-10.5 P-2ESect-10.5 P-3ESect-10.5 P-4ESect-10.5 P-5ESect-10.5 P-6ESect-10.5 P-7ESect-10.5 P-8ESect-10.5 P-9ESect-10.5 P-10ESect-10.5 P-11ESect-10.5 P-12ESect-10.5 P-13ESect-10.5 P-14ESect-10.5 P-15ESect-10.5 P-16ESect-10.5 P-17ESect-10.5 P-18ESect-10.5 P-19ESect-10.5 P-20ESect-10.5 P-21ESect-10.5 P-22ESect-10.5 P-23ESect-10.5 P-24ESect-10.5 P-25ESect-10.5 P-26ESect-10.5 P-27ESect-10.5 P-28ESect-10.5 P-29ESect-10.5 P-30ESect-10.5 P-31ESect-10.5 P-32ESect-10.5 P-33ESect-10.5 P-34ESect-10.5 P-35ESect-10.5 P-36ESect-10.5 P-37ESect-10.5 P-38ESect-10.5 P-39ESect-10.5 P-40ESect-10.5 P-41ESect-10.5 P-42ESect-10.5 P-43ESect-10.5 P-44ESect-10.5 P-45ESect-10.5 P-46ESect-10.5 P-47ESect-10.5 P-48ESect-10.5 P-49ESect-10.5 P-50ESect-10.6 P-1ESect-10.6 P-2ESect-10.6 P-3ESect-10.6 P-4ESect-10.6 P-5ESect-10.6 P-6ESect-10.6 P-7ESect-10.6 P-8ESect-10.6 P-9ESect-10.6 P-10ESect-10.6 P-11ESect-10.6 P-12ESect-10.6 P-13ESect-10.6 P-14ESect-10.6 P-15ESect-10.6 P-16ESect-10.6 P-17ESect-10.6 P-18ESect-10.6 P-19ESect-10.6 P-20ESect-10.6 P-21ESect-10.6 P-22ESect-10.6 P-23ESect-10.6 P-24ESect-10.6 P-25ESect-10.6 P-26ESect-10.6 P-27ESect-10.6 P-28ESect-10.6 P-29ESect-10.6 P-30ESect-10.6 P-31ESect-10.6 P-32ESect-10.6 P-33ESect-10.6 P-34ESect-10.6 P-35ESect-10.6 P-36ESect-10.6 P-37ESect-10.6 P-38ESect-10.6 P-39ESect-10.6 P-40ESect-10.6 P-41ESect-10.6 P-43ESect-10.6 P-44ESect-10.6 P-45ESect-10.6 P-46ESect-10.6 P-49ESect-10.6 P-50ESect-10.CR P-1CRSect-10.CR P-2CRSect-10.CR P-3CRSect-10.CR P-4CRSect-10.CR P-5CRSect-10.CR P-6CRSect-10.CR P-7CRSect-10.CR P-8CRSect-10.CR P-9CRSect-10.CR P-10CRSect-10.CR P-11CRSect-10.CR P-12CRSect-10.CR P-13CRSect-10.CR P-14CRSect-10.CR P-15CRSect-10.CR P-16CRSect-10.CR P-17CRSect-10.CR P-18CRSect-10.CR P-19CRSect-10.CR P-20CRSect-10.CR P-21CRSect-10.CR P-22CRSect-10.CR P-23CRSect-10.CR P-24CRSect-10.CR P-25CRSect-10.CR P-26CRSect-10.CR P-27CRSect-10.CR P-28CRSect-10.CR P-29CRSect-10.CR P-30CRSect-10.CR P-31CRSect-10.CR P-32CRSect-10.CR P-33CRSect-10.CR P-34CRSect-10.CR P-35CRSect-10.CR P-36CRSect-10.CR P-37CRSect-10.CR P-38CRSect-10.CR P-39CRSect-10.CR P-40CRSect-10.CT P-1CTSect-10.CT P-2CTSect-10.CT P-3CTSect-10.CT P-4CTSect-10.CT P-5CTSect-10.CT P-6CTSect-10.CT P-7CTSect-10.CT P-8CTSect-10.CT P-9CTSect-10.CT P-10CTSect-10.CT P-11CTSect-10.CT P-12CTSect-10.CT P-13CTSect-10.CT P-14CTSect-10.CT P-15CTSect-10.CT P-16CTSect-10.CT P-17CTSect-10.CT P-18CTSect-10.CT P-19CTSect-10.CT P-20CTSect-10.CT P-21CTSect-10.CT P-22CT

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