Question

which segment is adjacent to $\angle y$?
answer
$\overline{yf}$ $\overline{fm}$ $\overline{ym}$

Explanation:

To determine the segment adjacent to $\angle Y$, we recall that adjacent sides to an angle in a triangle share a common vertex with the angle and form the angle. $\angle Y$ is at vertex $Y$, so we look for segments that meet at $Y$ and form $\angle Y$. The segments meeting at $Y$ are $\overline{YF}$ and $\overline{YM}$. Among the options, $\overline{YF}$ is a side forming $\angle Y$ (the other is $\overline{YM}$, but we check the options). Wait, actually, in triangle $YFM$ (right - angled at $F$), the sides adjacent to $\angle Y$ are $\overline{YF}$ and $\overline{YM}$? Wait, no. Let's clarify: an adjacent segment to an angle is a segment that is one of the two sides forming the angle. So for $\angle Y$, the two sides are $\overline{YF}$ (from $Y$ to $F$) and $\overline{YM}$ (from $Y$ to $M$)? Wait, no, in the triangle, the vertices are $Y$, $F$, $M$. So the sides are $\overline{YF}$, $\overline{FM}$, and $\overline{YM}$. The angle at $Y$ is between $\overline{YF}$ and $\overline{YM}$? Wait, no, looking at the diagram, the right angle is at $F$, so the triangle is right - angled at $F$, so the sides are $FM$ (horizontal), $FY$ (vertical), and $YM$ (hypotenuse). So angle at $Y$ is between $FY$ and $YM$? Wait, no, the angle at $Y$: the two sides forming $\angle Y$ are $\overline{YF}$ (from $Y$ to $F$) and $\overline{YM}$ (from $Y$ to $M$)? Wait, no, $F$ is connected to $Y$ and $M$, $Y$ is connected to $F$ and $M$. So the angle at $Y$ is between $\overline{YF}$ and $\overline{YM}$. But among the options, $\overline{YF}$ is an option, $\overline{FM}$ is not adjacent (it's opposite to $\angle Y$? No, $\overline{FM}$ is between $F$ and $M$, not connected to $Y$ directly in terms of forming $\angle Y$). Wait, no, $\overline{FM}$ is a side from $F$ to $M$, so it doesn't share the vertex $Y$ to form $\angle Y$. So the segments adjacent to $\angle Y$ must share vertex $Y$ and be part of the angle. So $\overline{YF}$ is a side forming $\angle Y$ (the other is $\overline{YM}$, but $\overline{YF}$ is one of the sides adjacent to $\angle Y$.

Answer:

$\overline{YF}$