Is it possible for AB=10, BC=2x + 5 and AC=5x, given that point B is the midpoint? Why or why not?

Question

Is it possible for AB=10, BC=2x + 5 and AC=5x, given that point B is the midpoint? Why or why not?

Answer 1

OK:

If B is the midpoint, then AC = 2AB, 2BC, or AB + BC. AB also must = BC.

So, set it up as if the assumption is true.
1.) AB = 10 = BC = 2x +5
2.) AC = 5x = 2AB = 20 = 2BC = 4x + 10 = AB + BC = 2x +15

So: for 1, if 2x + 5 = 10:
2x = 5
x = 2.5

For 2: 5 x = 20, x = 4
or 5 x = 4x + 10
x = 10
or 5x = 2x + 15
3x = 15
x = 5

Since all the values of X differ once you've set the equation up, then no, it is not possible.

Answer 2

not possible. assuming you're talking about a straight line ABC,
there are 3 ways to work out x, but it doesn't meet the third criteria.

For example, with B as the midpoint, you need AB = BC, so 2x + 5 = 10. This gives x = 2.5.
In the other equation, this gives AC = 5x = 12.5, but the total length clearly needs to be 20 in this case.

Using the fact that the total length is 10 + 2x + 5 = 5x, this simplifies to 15 = 3x, and x = 5. This gives a total length of 25, which means B isn't in the middle, as AB = 10, BC = 15.

Finally, for B to be in the middle, need 2x(BC) = AC
this gives 2 x (2x + 5) = 5x. which leads to 4x + 10 = 5x, which simplifies to 10 = x, which means the total length is 50, AB = 10, and BC = 25 which doesn't even add up.

Therefore NOT POSSIBLE