# To show that the points A ( − 1 , 3 ) B ( 3 , 11 ) and C ( 5 , 15 ) are collinear by showing that | A B | + | B C | = | A C | ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter B, Problem 6E

(a)

To determine

## To calculate: To show that the points A(−1,3)  B(3,11) and C(5,15) are collinear by showing that |AB|+|BC|=|AC|

Expert Solution

It isproved that points are collinear by showing that |AB|+|BC|=|AC|

### Explanation of Solution

Given information: Points are A(1,3)

B(3,11) and C(5,15)

Formula Used:

Distance between two points is given as

D=(x2x1)2+(y2y1)2

Calculation:

Points are given as

A(1,3)

B(3,11) and C(5,15)

Length of AB is calculated as follows:

|AB|=(3(1))2+(113)2|AB|=42+82|AB|=16+64|AB|=80|AB|=45

Length of BC is calculated as follows:

|BC|=(53)2+(1511)2|BC|=22+42|BC|=4+16|BC|=20|BC|=25

Length of AC is calculated as follows:

|AC|=(5(1))2+(153)2|AC|=62+122|AC|=36+144|AC|=180|AC|=65

Now, calculating |AB|+|BC| ,

|AB|+|BC|=45+25|AB|+|BC|=65|AB|+|BC|=|AC|

Thus, points A, B and C are collinear

Conclusion:

Hence, its proved that points are collinear by showing that |AB|+|BC|=|AC|

(b)

To determine

### To calculate: To show that the points A(−1,3)  B(3,11) and C(5,15) are collinear by using slopes

Expert Solution

It isproved that points are collinear by showing that slopes are equal

### Explanation of Solution

Given information: Points are A(1,3)

B(3,11) and C(5,15)

Formula Used:

Slope of line joining two points is calculated as

m=y2y1x2x1

Calculation:

Points are given as

A(1,3)

B(3,11) and C(5,15)

Slope of line AB is calculated as follows:

mAB=1133(1)mAB=84mAB=2

Slope of line BC is calculated as follows:

mBC=151153mBC=42mBC=2

Slope of line AC is calculated as follows:

mAC=1535(1)mAC=126mAC=2

Since mAB=mBC=mAC=2

Thus, points A, B and C are collinear

Conclusion:

Hence, its proved that points are collinear by showing that slope of lines are equal

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