Question

find vw.
vw =
submit

Explanation:

Step1: Identify the theorem

Since $XT = 66 - 22 = 44$? Wait, no, the two lines with ticks are parallel? Wait, the marks on the sides: the two black lines with ticks are equal? Wait, no, the blue lines: $TW$ and $WV$ have arrows, so $TW \parallel WV$? Wait, actually, the triangle has a midline? Wait, no, let's check the lengths. The segment $UT$ is 22, and the other side (the black one) is 66. So the ratio of $UT$ to the whole side: $UT = 22$, and the total length from $U$ to the top (the black line) is 66? Wait, no, the black line is a side with length 66, and $UT$ is 22. So the ratio of $UT$ to the entire side (let's say $UX$? Wait, no, the triangle: $XU$ is a side, with $T$ on $XU$? Wait, maybe it's the midline theorem. Wait, the segment $TW$ is parallel to $XV$? No, wait, the arrows on $TW$ and $WV$? Wait, no, the arrows on $TW$ and $VV$? Wait, the diagram: $XW = 50$, $UT = 22$, and the other side (the black one) is 66. So $UT = 22$, and the length from $U$ to the vertex (let's say $X$) is 66? Wait, no, the black line is a side with length 66, and $UT$ is 22. So the ratio of $UT$ to the entire side: $22/66 = 1/3$. Wait, maybe $TW$ is a midline? No, the midline theorem says that the midline is parallel to the base and half its length. But here, the ratio is 22/66 = 1/3. So if $UT$ is 22 and the total length is 66, then the ratio of similarity is 1/3? Wait, no, maybe $T$ divides $XU$ into $XT$ and $TU$, with $TU = 22$ and $XT = 66 - 22 = 44$? No, that doesn't make sense. Wait, the two black lines with ticks: they are equal, so the sides are equal? Wait, no, the black lines are on the left, with ticks, so they are congruent. So the length of the left side is 66, and $UT$ is 22. So the ratio of $UT$ to the left side is 22/66 = 1/3. Then, by the basic proportionality theorem (Thales' theorem), if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. So if $TW$ is parallel to $XV$ (wait, no, $XW$ is 50, and we need to find $VW$. Wait, let's denote: let’s say triangle $XUV$, with $T$ on $XU$ and $W$ on $XV$, such that $TW \parallel UV$? No, the arrows on $TW$ and $WV$? Wait, maybe the ratio of $UT$ to $XU$ is 22/66 = 1/3. Then, by the proportionality theorem, $VW / XW = UT / XU$. Wait, $XW = 50$, so $VW = XW \times (UT / XU) = 50 \times (22/66) = 50 \times (1/3) = 50/3$. Wait, that makes sense. So the ratio of $UT$ to the entire side (66) is 22/66 = 1/3, so $VW$ is 1/3 of $XW$? Wait, no, maybe $XW$ and $VW$ are segments on $XV$, and $TW$ is parallel to $XU$? No, the diagram is a bit unclear, but the key is the ratio. Since $UT = 22$ and the total length (the side with length 66) gives a ratio of 22/66 = 1/3, so $VW = 50 \times (1/3) = 50/3$.

Step2: Calculate the ratio

The length of $UT$ is 22, and the length of the entire side (let's say $XU$) is 66. So the ratio of $UT$ to $XU$ is $\frac{22}{66} = \frac{1}{3}$.

Step3: Apply the ratio to $XW$

Since the triangles are similar (by the basic proportionality theorem or similar triangles), the ratio of $VW$ to $XW$ is the same as the ratio of $UT$ to $XU$. So $VW = XW \times \frac{1}{3} = 50 \times \frac{1}{3} = \frac{50}{3}$.

Answer:

$\frac{50}{3}$