SAT Lines

Correct Answer: E

Solution:

Tip: The midpoint of a segment divides it in half.
We show why all of the answer choices are true, except choice (E):

(A) If $M$ is the midpoint of segment $\overline{AB},$ then $AM= MB = \frac{1}{2}AB.$ Therefore this statement is true.

(B) Recall that if $m$ and $n$ are on a number line, and $m<n$ , then the distance between $m$ and $n$ is $n-m.$ In this case, $b>x$ and therefore $MB = b-x.$ This statement is true.

(C) By definition, the midpoint divides a segment into two congruent segments. If $M$ is the midpoint of $\overline{AB},$ as is the case here, then $\overline{AM} \cong \overline{MB}.$

(D) $AM = a-x$ and $BM = b-x.$ By the definition of a midpoint, $\overline{AM} \cong \overline{MB}$ or $AM = MB.$ It follows then that $a-x = b-x,$ and re-arranging, we obtain $2x =a+b.$ Therefore (D) is also true.

(E) Following the reasoning in (D), we conclude that $x = \frac{a+b}{2} \neq 2b-a.$ Option (E) is false, and it is the correct answer.

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Incorrect Choices:

(A) , (B) , (C) , and (D)
The solution explains why these choices are wrong.