We are given a diagram with a circle inscribed in a triangle. We are given the lengths $QR = 24$ ft, $UV = 26$ ft, and $US = 36$ ft. We need to find the length of $QU$.

1. Problem Description

We are given a diagram with a circle inscribed in a triangle. We are given the lengths $QR = 24$ ft, $UV = 26$ ft, and $US = 36$ ft. We need to find the length of $QU$.

2. Solution Steps

Since tangents from a point to a circle have equal lengths, we have:

$QR = QV = 24$

$US = UT = 36$

$VU = US + SV$

$VT = VU = 26$

Since $VU$ and $QU$ are tangents to the circle from point $V$ and $U$, we have

$VU = 26$.

Now, $VU = VT + TU$.

Thus, $UT = US - TS$.

Since the two tangents from an exterior point to a circle are equal, we have $RT = TS$.

So, $QT = RS = x$.

$RS = US = 36$

Since $QS$ consists of $QR$ and $RS$, and $QU$ consists of $QV$ and $VU$, $RU = RV$.

Also, since two tangents to the same circle from the same exterior point have equal lengths, then

$QR= QV=24$

$TS= RS$

$US = UT = 36$ and $UV = VT = 26$

We also have $US = UT + TS = 36$

Since we know that $QU = QV+VU = 24+26=50$.

$US = UT$ so $VT = 26$.

$QU=QV+VU=24+VU= QV + 26=50$

Since $UR$ and $UR$ are tangent segments from $U$, $UR=UT$. Also $UR=UV = x$.

Since $VR$ and $VR$ are tangent segments from $V$, $VR=VS = y$.

Then

$QR = QV+RV$, so $24=x+26 \Rightarrow x = -2$, so this assumption is wrong.

We have $US = 36$, $VU = 26$, $QR = 24$.

We have $QU = QV + VU$. Also $QV = QR$, so $QU = QR + VU$ should not be equal to the line segment QV = length of VT+ TU because we don't even know where those lengths lie on the figure.

If we consider $QR =24$ and $QU = QV+ VR +RU= 24 ft + VU$.

Let x= TS. Then RU = x. US = UT + TS and we also know that US = 36

Since UT = RU, this means x+UT=36 ft

VT = UV and TS = SR. We also know that SR + QR = QS and VT +QV =VU

ST+QR

Since tangent segments to a circle have the same length. Then UV=x and RS = Y . Let US=36 . And Let the value for RT be w. So. RT = TS =w

$QT = \sqrt{QU^2}$

Based on the figure, we cannot deduce a specific length for $QU$. However, based on the previous answers, we may want to analyze triangle $QVU$.

3. Final Answer

Cannot determine the length of $QU$ with the given information.