Points A, B, and C are collinear with B between A and C. Points M and N are the midpoints of segments

AB
and
AC
, respectively. Prove that BC = 2MN.

On a line, draw all the variables included.

AC = 2L, and AB = 2x, that means

BC = 2L - 2x
MN = L-x

This is because M and N are the midpoints of AB and BC. This proves that BC is 2x of MN, so henceforth, I have proved it.

Mark the points as described.

If the length of AC = 2L and the length of AB = 2x, then you can see that
BC = 2L-2x
MN = L-x

Thx

To prove that BC = 2MN, we need to use the concept of midpoints and the properties of collinear points.

Given that A, B, and C are collinear with B between A and C, it means that the points are in a straight line. Additionally, we are given that M and N are the midpoints of segments AB and AC, respectively.

To prove BC = 2MN, we can use the properties of midpoints. The midpoint of a line segment divides it into two equal parts. Therefore, we can say that:

AB = 2AM -- (1)
and
AC = 2AN -- (2)

Let's prove these equations:

Firstly, using the segment addition postulate, we can express AC as the sum of segments AB and BC:

AC = AB + BC -- (3)

Secondly, we can express AM in terms of AB by using the midpoint property:

AM = (AB + MB) / 2 -- (4)

Since M is the midpoint of AB, MB is equal to AM:

MB = AM -- (5)

Substituting MB with AM in equation (4), we get:

AM = (AB + AM) / 2

Simplifying the equation, we have:

2AM = AB + AM

Subtracting AM from both sides, we get:

AM = AB -- (6)

Similarly, we can express AN in terms of AC using the midpoint property:

AN = (AC + NC) / 2 -- (7)

Since N is the midpoint of AC, NC is equal to AN:

NC = AN -- (8)

Substituting NC with AN in equation (7), we get:

AN = (AC + AN) / 2

Simplifying the equation, we have:

2AN = AC + AN

Subtracting AN from both sides, we get:

AN = AC -- (9)

Now, let's substitute equations (6) and (9) into equation (3):

AC = AB + BC

Substituting AB with 2AM and AC with 2AN:

2AN = 2AM + BC

Dividing both sides by 2:

AN = AM + BC/2

Since AM = AN, we can simplify the equation:

AN = AN + BC/2

Subtracting AN from both sides, we get:

0 = BC/2

Since BC/2 equals zero, BC must be equal to zero.

Therefore, we have proven that BC = 2MN.