the perimeter of a right triangle ABC is 36 cm A circle is inscribed in the triangle which divides - Maths - Circles

Question

the perimeter of a right triangle ABC is 36 cm A circle is
inscribed in the triangle which divides the hypotenuse in thr ratio
of 2:3 What are the lenghts of the sides of the triangle?

Rishabh Mittal · Accepted Answer

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As hypotenuse is divided in 2:3 
So, Let AP=2x and PC=3x 
AP=AR=2x     Tangents drawn from same external point to circle are equalCP=CQ=3x     Tangents drawn from same external point to circle are equalLet BR=ySo,BR=BQ=y       Tangents drawn from same external point to circle are equalNow , Perimeter=36 cm2x+3x+2x+y+3x+y=3610x+2y=365x+y=18y=18-5xNow,According to Pythagoras theoremAC2=AB2+BC25x2=2x+y2+3x+y225x2=2x+18-5x2+3x+18-5x225x2=18-3x2+18-2x225x2=324+9x2-108x+324+4x2-72x12x2+180x-648=0x2+15x-54=0x2+18x-3x-54=0xx+18-3x+18=0x-3 x+18 =0x=3 and x=-18Length can't be negative 
So, x=3 cm
Now,AB=2x+18-5x=18-3x=18-9=9 cmBC=3x+18-5x=18-2x=18-6=12 cm 
AC=2x+3x=5x=15 cm

the perimeter of a right triangle ABC is 36 cm. A circle is inscribed in the triangle which divides the hypotenuse in thr ratio of 2:3. What are the lenghts of the sides of the triangle?