Question

9. Given the circle below, find OR.

Answer 1

A. 17

Explanation:

Let X be the intersection of lines QR and NP.

By Power of a Point, we have
`QX\cdot XR=NX\cdot XP.`

(This can be proven with similar triangles by connecting QN and PR and using proportions with similar triangles
`\triangle QXN` and
`\triangle RXP`.)

Plugging the values we know into
`QX\cdot XR=NX\cdot XP,` we have

`12\cdot(x-2)=(3x-1)\cdot 3`.

Dividing both sides by 3 gives

`4(x-2)=3x-1.`

Distributing the left hand side gives

`4x-8=3x-1.`

Subtracting 3x from both sides, we have

`x-8=-1.`

Finally, adding 8 to both sides, we have

`x=7.`

The question asks to find QR. (I'm assuming this is what you meant in the question.) With the lengths given, we know

`QR=12+(x-2)=x+10.`

Therefore, plugging x in gives us

`\boxed{QR=17.}`