We are given two lengths of the rectangle:
RU=3x-6
UT=x+9
These two lengths are shown in the following diagram:
Since this is a rectangle, the lengths of RU and UT must be equal:
[tex]RU=UT[/tex]
Thus
[tex]3x-6=x+9[/tex]
We need to solve this equation for x.
We start by subtracting x to both sides of the equation:
[tex]\begin{gathered} 3x-x-6=9 \\ 2x-6=9 \end{gathered}[/tex]
Now, add 6 to both sides:
[tex]\begin{gathered} 2x=9+6 \\ 2x=15 \end{gathered}[/tex]
Finally, divide both sides by 2:
[tex]\begin{gathered} \frac{2x}{2}=\frac{15}{2} \\ x=7.5 \end{gathered}[/tex]
We have the value of x: x=7.5
Now we have to find the length of QS. Since QS and RT are diagonals of the same rectangle, they have to be equal:
[tex]RT=QS[/tex]
This means that we can find RT by adding RU and UT, and the result will be equal to QS:
[tex]QS=RU+TU[/tex]
substituting the given expressions for RU and TU:
[tex]QS=3x-6+x+9[/tex]
And now, substitute x=7.5 and solve for QS:
[tex]QS=3(7.5)-6+7.5+9[/tex][tex]\begin{gathered} QS=22.5-6+7.5+9 \\ QS=33 \end{gathered}[/tex]
Answer:
x=7.5 and QS=33
