# To verify: The points A ( − 1 , 3 ) , B ( 3 , 11 ) and C ( 5 , 15 ) are collinear by showing that d ( A , B ) + d ( B , C ) = d ( A , C ) .

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.8, Problem 42E
To determine

## To verify: The points A(−1,3),B(3,11) and C(5,15) are collinear by showing that d(A,B)+d(B,C)=d(A,C) .

Expert Solution

### Explanation of Solution

Given information:

The points A(1,3),B(3,11) and C(5,15) and the equation d(A,B)+d(B,C)=d(A,C) .

Formula used:

Distance formula between two points X(a1,b1) and Y(a2,b2) is calculated as,

d(XY)=(a2a1)2+(b2b1)2

Proof:

Consider the given equation d(A,B)+d(B,C)=d(A,C) , it can be proved by solving left hand side and right hand side separately.

Recall that the distance formula between two points X(a1,b1) and Y(a2,b2) is calculated as,

d(XY)=(a2a1)2+(b2b1)2

So, left hand side of the equation will be solved as,

d(A,B)+d(B,C)=(3(1))2+(113)2+(53)2+(1511)2=(4)2+(8)2+(2)2+(4)2=16+64+4+16

Simplify it further as,

d(A,B)+d(B,C)=16+64+4+16=80+20=45+25=65

Now, solve the right hand side of the equation as,

d(A,C)=(15)2+(315)2=(6)2+(12)2=36+144

Simplify it further as,

d(A,C)=36+144=180=65

Since, left hand side and right hand side are equal, therefore, it is proved that the given points A(1,3),B(3,11) and C(5,15) are collinear as d(A,B)+d(B,C)=d(A,C) .

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