Question

Beatrice calculated the slope between two pairs of points. She found that the slope between ( ( -3, 3 , -2) 2 ) and (1, 0) ( 1 , 0 ) is 12 1 2 . She also found that the slope between ( ( -2, 2 , -1) 1 ) and (4, 2) ( 4 , 2 ) is 12 1 2 . Beatrice concluded that all of these points are on the same line. Use the drop-down menus to complete the statements about Beatrice's conclusion

Answer 1

These points may or may not be on the same line.

Explanation:

The first set of points is: (-3, -2) and (1, 0)

The second set of points is (-2, -1) and (4,2)

Slope of first two points is:

`Slope=(0-(-2))/(1-(-3))=(2)/(4)=(1)/(2)`

Slope of second set of points is:

`Slope=(2-(-1))/(4-(-2))=(3)/(6)=(1)/(2)`

The points are on the lines with the same slopes. But this does not guarantee that these points are on the same line as two parallel lines will also have the same slopes. In order to confirm the conclusion Beatrice need to find the y-intercept or the equation of the lines. If the y-intercept or the equation of lines is same in both cases this would mean that all 4 points are on the same line, else they would be on 2 different lines.

Answer 2

Answer with explanation:

The Slope between two points (-3,-2) and (1,0) is equal to
`(1)/(2)`

Also, Slope between two points (-2,-1) and (4,2) is equal to
`(1)/(2)`

→To check whether ,all these points lie on the same line or not,just plot these points on the two dimensional plane and check whether they are collinear or not.You will find that ,these segments are parallel.

→The second method is, find area of triangle using any three ordered pair.if area of triangle is zero ,it means that these three points are collinear.

→Formula for Area of triangle having coordinates ,(a,b),(p,q) and (r,s).

`=(1)/(2)[a(q-s)-p(s-b)+r(b-q)]`

→If two segments have same slope , it means they are parallel or may be coincident also.

Finding the equation of line passing through two points, (a,b) and (p,q).

y-b=m(x-a), where ,
`m=(b-q)/(a-p)`

⇒Equation of line passing through (-3,-2) and (1,0), and having slope
`(1)/(2)` is

→2×(y-0)=1×(x-1)

2 y-x+1=0

→x=2 y+1

⇒Equation of line passing through (-2,-1) and (4,2), and having slope
`(1)/(2)` is

→2×(y-2)=1×(x-4)

2 y-4-x+4=0

2 y-x=0

→x=2 y

The two lines, have same slope but ,different Y intercept,that is 1 and 0,means lines are parallel not coincident.

So, Beatrice concluded that all of these points are on the same line is false statement.