Question

Help plsWhat is the length of RP? a) 3b) 8c) 5d) not here

Answer 1

The locus of all points from which a given segment subtends equal angles is a circle. Therefore:

`RT\cdot RS=RQ\cdot RP`

so:

`\begin{gathered} (4+6)(4)=(x+2+x)(x+2) \\ 10(4)=(2x+2)(x+2) \\ 40=2x^2+6x+4 \\ \end{gathered}`

Divide both sides by 2:

`\begin{gathered} x^2+3x+2=20 \\ so\colon \\ x^2+3x-18=0 \end{gathered}`

The factors of -18 that sum to 3 are 6 and -3, therefore:

`\begin{gathered} x^2+3x-18=(x+6)(x-3) \\ so\colon \\ x=3 \\ or \\ x=-6 \end{gathered}`

So:

`\begin{gathered} x=3 \\ because\colon \\ RP>0 \\ RP=x+2 \\ RP=3+2 \\ RP=5 \end{gathered}`