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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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In Exercises 5 to 10, the real numbers a, b, c, and d are positive.

Consider the triangle with vertices at A 0 ,   0 , B a ,   0 , and C a ,   b . Explain why A B C is a right triangle.

Chapter 10.3, Problem 5E, In Exercises 5 to 10, the real numbers a, b, c, and d are positive. Consider the triangle with

To determine

To explain:

ABC is a right triangle.

Explanation

The given figure is shown below.

To prove ABC is a right triangle, we have to prove the following statement.

The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices of the triangle.

The given vertices are A0, 0, Ba, 0, and Ca, b.

The endpoints of the hypotenuse line are A0, 0 and Ca, b.

Find the midpoint of the hypotenuse line.

x1,y1=0, 0

x2,y2=a, b

Formula for midpoint of the line segment joining two points.

M=x1+x22, y1+y22

Substituting the x and y co-ordinates.

M=0+a2, 0+b2

M=a2, b2

Let this midpoint be named as Da2, b2.

Now, find the distance between this midpoint D and A.

x1,y1=Da2, b2

x2,y2=A0, 0

Formula for distance between two points

dDA=x2-x12+y2-y12

Substituting the x and y co-ordinates

dDA=0-a22+0-b22

On solving this,

dDA=-a22+-b22

Squaring,

dDA=a24+b24

dDA=a2+b24

Taking square root,

dDA=a2+b24

dDA=a2+b22

Now, find the distance between this midpoint D and B

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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-10.2 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-10.3 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-10.4 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-27ESect-10.4 P-28ESect-10.4 P-29ESect-10.4 P-30ESect-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-10.6 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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