Find up to three places of …

We have to find  upto  three places of decimal the radius of the circle whose area is the sum of the areas of two triangles whose sides are 35, 53, 66 and 33, 56, 65 measured in centimetres. {tex}{/tex} 

For the first triangle, we have a = 35, b = 53 and c = 66.
{tex}\therefore \quad s = \frac { a + b + c } { 2 } = \frac { 35 + 53 + 66 } { 2 } = 77 \mathrm { cm }{/tex}
Let {tex}\Delta _ { 1 }{/tex} be the area of the first triangle. Then,
{tex}\Delta _ { 1 } = \sqrt { s ( s - a ) ( s - b ) ( s - c ) }{/tex}
{tex}\Rightarrow \quad \Delta _ { 1 } = \sqrt { 77 ( 77 - 35 ) ( 77 - 53 ) ( 77 - 66 ) } = \sqrt { 77 \times 42 \times 24 \times 11 }{/tex}
{tex}\Rightarrow \quad \Delta _ { 1 } = \sqrt { 7 \times 11 \times 7 \times 6 \times 6 \times 4 \times 11 }{/tex}= {tex}\sqrt { 7 ^ { 2 } \times 11 ^ { 2 } \times 6 ^ { 2 } \times 2 ^ { 2 } } = 7 \times 11 \times 6 \times 2 = 924 \mathrm { cm } ^ { 2 }{/tex} ...(i)
For the second triangle, we have a = 33,b = 56,c = 65
{tex}\therefore \quad s = \frac { a + b + c } { 2 } = \frac { 33 + 56 + 65 } { 2 } = 77 \mathrm { cm }{/tex}
Let {tex}\Delta{/tex}₂ be the area of the second triangle. Then,
{tex}\Delta _ { 2 } = \sqrt { s ( s - a ) ( s - b ) ( s - c ) }{/tex}
{tex}\Rightarrow \quad \Delta _ { 2 } = \sqrt { 77 ( 77 - 33 ) ( 77 - 56 ) ( 77 - 65 ) }{/tex}
{tex}\Rightarrow \quad \Delta _ { 2 } = \sqrt { 77 \times 44 \times 21 \times 12 }{/tex}= {tex}\sqrt { 7 \times 11 \times 4 \times 11 \times 3 \times 7 \times 3 \times 4 } = \sqrt { 7 ^ { 2 } \times 11 ^ { 2 } \times 4 ^ { 2 } \times 3 ^ { 2 } }{/tex}
{tex}\Rightarrow \quad \Delta _ { 2 } = 7 \times 11 \times 4 \times 3 = 924 \mathrm { cm } ^ { 2 }{/tex}
Let r be the radius of the circle. Then,
Area of the circle = Sum of the areas of two triangles
{tex}\Rightarrow \quad \pi r ^ { 2 } = \Delta _ { 1 } + \Delta _ { 2 }{/tex}
{tex}\Rightarrow \quad \pi r ^ { 2 }{/tex}= 924 + 924
{tex}\Rightarrow \quad \frac { 22 } { 7 } \times r ^ { 2 } = 1848{/tex}
{tex}\Rightarrow \quad r ^ { 2 } = 1848 \times \frac { 7 } { 22 } = 3 \times 4 \times 7 \times 7 \Rightarrow{/tex}{tex}r = \sqrt { 3 \times 2 ^ { 2 } \times 7 ^ { 2 } } = 2 \times 7 \times \sqrt { 3 } = 14 \sqrt { 3 } \mathrm { cm }{/tex}