Given:
The figure of triangle GHI and a circle M inscribed in the triangle.
To find:
The perimeter of the triangle.
Solution:
We know that the lengths of the tangent on a circle from same exterior point are always equal.
[tex]JH=HK[/tex] (Tangent from point H)
[tex]6x-11=2x+9[/tex]
[tex]6x-2x=11+9[/tex]
[tex]4x=20[/tex]
Divide both sides by 4.
[tex]x=\dfrac{20}{4}[/tex]
[tex]x=5[/tex]
Now,
[tex]JH=6x-11[/tex]
[tex]JH=6(5)-11[/tex]
[tex]JH=30-11[/tex]
[tex]JH=19[/tex]
In the same way,
[tex]GJ=GL[/tex] (Tangent from point H)
[tex]IK=IL[/tex] (Tangent from point H)
From the figure, it is clear that,
[tex]IL=GI-GL[/tex]
[tex]IL=40-GJ[/tex]
[tex]IL=40-23[/tex]
[tex]IL=17[/tex]
The perimeter of the triangle GHI is:
[tex]Perimeter=GH+HI+GI[/tex]
[tex]Perimeter=(GJ+HJ)+(HK+IK)+(GL+IL)[/tex]
[tex]Perimeter=(23+19)+(19+17)+(23+17)[/tex]
[tex]Perimeter=118[/tex]
Therefore, the perimeter of the triangle GHI is 118 units.
