Answer:
A. 17
Step-by-step explanation:
Let X be the intersection of lines QR and NP.
By Power of a Point, we have [tex]QX\cdot XR=NX\cdot XP.[/tex]
(This can be proven with similar triangles by connecting QN and PR and using proportions with similar triangles [tex]\triangle QXN[/tex] and [tex]\triangle RXP[/tex].)
Plugging the values we know into [tex]QX\cdot XR=NX\cdot XP,[/tex] we have
[tex]12\cdot(x-2)=(3x-1)\cdot 3[/tex].
Dividing both sides by 3 gives
[tex]4(x-2)=3x-1.[/tex]
Distributing the left hand side gives
[tex]4x-8=3x-1.[/tex]
Subtracting 3x from both sides, we have
[tex]x-8=-1.[/tex]
Finally, adding 8 to both sides, we have
[tex]x=7.[/tex]
The question asks to find QR. (I'm assuming this is what you meant in the question.) With the lengths given, we know
[tex]QR=12+(x-2)=x+10.[/tex]
Therefore, plugging x in gives us
[tex]\boxed{QR=17.}[/tex]
