The radius is equal to 5 cm.
Given:
OP=x-6, PT=x+1, OT=x+2
To find:
The radius
Solution:
The radius OP is perpendicular to the tangent PT.
Angle OPT=90°
In ΔOPT, using the Pythagoras theorem,
[tex]OP^{2} +PT^{2} =OT^{2}[/tex]
Using values,
[tex](x-6)^{2}[/tex]+[tex](x+1)^{2}[/tex]=[tex](x+2)^{2}[/tex]
[tex]x^{2}[/tex]+36-12x+[tex]x^{2}[/tex]+1+2x = [tex]x^{2}[/tex]+4+4x
2[tex]x^{2}[/tex]-10x+37=[tex]x^{2}[/tex]+4x+4
2[tex]x^{2}[/tex]-[tex]x^{2}[/tex]=4x+10x+4-37
[tex]x^{2}[/tex]=14x-33
[tex]x^{2}[/tex]-14x+33=0
On factorizing,
[tex]x^{2}[/tex]-11x-3x+33=0
x(x-11)-3(x-11)=0
(x-11)(x-3)=0
So, the values of x are-
x-11=0, x-3=0
x=11, x=3
Now, the triangle's sides have to be positive, so the value of x=11.
The circle's radius=OP=(x-6)
=11-6
=5 cm
Therefore, the radius is equal to 5 cm.