# The slope of the line that contains ( 5 , − 12 ) and ( − 7 , 4 ) .

### 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 T, Problem 4BDT

(a)

To determine

## To find: The slope of the line that contains (5,−12) and (−7,4).

Expert Solution

The slope of the line is m=43.

### Explanation of Solution

Formula used:

The slope of the line is m=y1y2x1x2, where the line passes through the points (x1,y1) and (x2,y2).

Calculation:

It is given that the line that contains (5,12) and (7,4).

Substitute x1=5, y1=12, x2=7, and y2=4 in m=y1y2x1x2 to find the slope of the line,

m=1245(7)=1612=43

Thus, the slope of the line is m=43.

(b)

To determine

### To find: The equation and the intercept of the line that contains (5,−12) and (−7,4).

Expert Solution

The intercept of the line is 163 and the equation of the line is y=43x163.

### Explanation of Solution

Given:

The slope of the line is m=43.

Formula used:

The equation of the straight line is y=mx+c, where m is the slope of the line, c is the x-intercept.

Calculation:

Suppose, the equation of the straight line is y=mx+c.

It is given that, the slope of the line is m=43.

Thus, substitute m=43 in y=mx+c to obtain the equation of the straight line,

y=43x+c

It is given that, the line passes through the point (7,4).

Thus, substitute x=7 and y=4 in y=43x+c to obtain the intercept of the straight line,

4=43(7)+cc=4283c=163

Thus, the value is c=163, that is the intercept of the line.

Thus, substitute c=163 in y=43x+c to obtain the equation of the straight line,

y=43x163

Thus, the intercept of the line is 163 and the equation of the line is y=43x163.

(c)

To determine

### To find: The mid points of the points (5,−12) and (−7,4).

Expert Solution

The midpoint of the points (5,12) and (7,4) is (1,4).

### Explanation of Solution

Formula used:

The midpoint of the points (x1,y1) and (x2,y2) is (x1x22,y1y22).

Calculation:

It is given that the points are (5,12) and (7,4).

Substitute x1=5, y1=12, x2=7, and y2=4 in (x1x22,y1y22) to find the midpoints,

(7+52,4122)=(22,82)=(1,4)

Thus, the midpoint of the points (5,12) and (7,4) is (1,4).

(d)

To determine

### To find: The length of the segment between the points (5,−12) and (−7,4).

Expert Solution

The length of the segment between the points (5,12) and (7,4) is 20.

### Explanation of Solution

Formula used:

The length of the segment between the points (x1,y1) and (x2,y2) is (x1x2)2+(y1y2)2.

Calculation:

It is given that the points are (5,12) and (7,4).

Substitute x1=5, y1=12, x2=7, and y2=4 in (x1x2)2+(y1y2)2 to find the midpoints,

(75)2+(4+12)2=(12)2+(16)2=144+256=400=±20

Length cannot be any negative term. Thus, the length is 20.

Thus, the length of the segment between the points (5,12) and (7,4) is 20.

(e)

To determine

### To find: The equation of the perpendicular bisector of AB. (5,−12) and (−7,4).

Expert Solution

The equation of the perpendicular bisector is y=34x134.

### Explanation of Solution

Given:

The points are A=(5,12) and B=(7,4).

The slope of the line AB is m=43.

The midpoint of the points (5,12) and (7,4) is (1,4).

Formula used:

The equation of the straight line is y=mx+c, where m is the slope of the line, c is the x-intercept.

Calculation:

Suppose, the equation of the straight line is y=mx+c.

It is given that, the slope of the line AB is 43.

Thus, slope of the perpendicular bisector of AB is 34.

Thus, substitute m=34 in y=mx+c to obtain the equation of the straight line,

y=34x+c

It is given that, the line passes through the midpoint, that is (1,4).

Thus, substitute x=1 and y=4 in y=34x+c to obtain the intercept of the straight line,

4=34(1)+cc=4+34c=134

Thus, the value is c=134, that is the intercept of the line.

Thus, substitute c=134 in y=34x+c to obtain the equation of the straight line,

y=34x134

Thus, the equation of the perpendicular bisector is y=34x134.

(f)

To determine

### To find: The equation of the circle that has diameter as AB.

Expert Solution

The equation of the circle is (x+1)2+(y+4)2=100.

### Explanation of Solution

Given:

The points are A=(5,12) and B=(7,4).

The midpoint of the points (5,12) and (7,4) is (1,4).

The length of the segment between the points (5,12) and (7,4) is 20.

Formula used:

The equation of the circle is (xh)2+(yk)2=r2, where the center of the circle is at the point (h,k), r is the radius of the circle.

Calculation:

Suppose, the equation of the circle is (xh)2+(yk)2=r2.

It is given that, the center of the circle is at the midpoint of the points (5,12) and (7,4), that is (1,4).

Thus, substitute h=1 and k=4 in (xh)2+(yk)2=r2 to obtain the equation of the straight line,

(x+1)2+(y+4)2=r2

The radius is the half of the length of the segment AB.

It is given that the length of the segment between the points (5,12) and (7,4) is 20.