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.