Given points A(2, 5) and B(5, 0) on a coordinate plane, determine the midpoint of segment AB.

Step-by-step solution

1. 1

   Recall the midpoint formula. The midpoint of a segment is the coordinatewise average of the endpoints:

   $M = \left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)$

   It is an average, not a difference — differences belong to the distance and slope formulas.

2. 2

   Average the $x$-coordinates.

   $x_M = \frac{2+5}{2} = \frac{7}{2} = 3.5$

3. 3

   Average the $y$-coordinates. The zero still counts as a value and must be included in the sum:

   $y_M = \frac{5+0}{2} = \frac{5}{2} = 2.5$

4. 4

   Write the midpoint.

   $M = (3.5,\ 2.5)$

5. 5

   Check that $M$ really is halfway. From $A(2,5)$ to $M$: $\Delta x = +1.5$, $\Delta y = -2.5$. From $M$ to $B(5,0)$: $\Delta x = +1.5$, $\Delta y = -2.5$ ✓ — identical steps, so $M$ is on segment $AB$ and equidistant from both ends. The distances also match: $\sqrt{1.5^2+2.5^2} = \sqrt{8.5} \approx 2.9155$ on each side, exactly half of $|AB| = \sqrt{3^2+5^2} = \sqrt{34} \approx 5.8310$.

6. 6

   Rule out the distractors. $(3.5, 0)$ averages only $x$ and copies $y$ from $B$; $(3, 5)$ mixes a difference with a copied coordinate; $(7, 5)$ is the sum of the coordinates with no halving at all.