We are given a triangle $PQR$ with a line segment $YX$ parallel to $QR$. The length of $PY$ is 2, the length of $YR$ is 3, and the length of $YX$ is 4. We need to find a triangle similar to triangle $PYX$ and determine the length of $QR$.

1. Problem Description

We are given a triangle $PQR$ with a line segment $YX$ parallel to $QR$. The length of $PY$ is 2, the length of $YR$ is 3, and the length of $YX$ is

4. We need to find a triangle similar to triangle $PYX$ and determine the length of $QR$.

2. Solution Steps

First, since $YX$ is parallel to $QR$, we have that $\angle PYX = \angle PQR$ and $\angle PXY = \angle PRQ$. Thus, by the Angle-Angle (AA) similarity criterion, triangle $PYX$ is similar to triangle $PQR$.

To find the length of $QR$, we can set up a proportion using the corresponding sides of the similar triangles $PYX$ and $PQR$. We have $PY/PR = YX/QR$.

We know $PY = 2$ and $YR = 3$, so $PR = PY + YR = 2 + 3 = 5$.

We are also given $YX = 4$.

Therefore, we have the proportion:

$2/5 = 4/QR$

Now, we solve for $QR$:

$2 \cdot QR = 5 \cdot 4$

$2 \cdot QR = 20$

$QR = 20/2$

$QR = 10$

3. Final Answer

Triangle $PYX$ is similar to triangle $PQR$.

The length of $QR$ is 10.