Question

Refer to the image?!

Prove:Area of the triangle`=R*S`;
where,
`R=`radius of the incircle,
and,
`S=`Semi-perimeter of the triangle,
which is,Perimeter of the triangle`/2`

Answer 1

`"see explanation"`

Explanation:

`"this is going to be 'tricky' without diagrams but here goes"`

`"from the centre of the circle draw line segments to each"`
`"vertex of the triangle"`

`"this 'splits' the main triangle into 3 triangles "`

`"draw in the 2 remaining radii to the point of contact with"`
`"the sides of the triangle similar to the one shown in the image"`

`"the sum of the areas of the 3 triangles are equal to"`
`"the area of the main triangle"`

`"let the 3 sides be "b_1,b_2" and "b_3`

`rArr"area "=1/2Rb_1+1/2Rb_2+1/2Rb_3`

`color(white)(rArr"area ")=1/2R(b_1+b_2+b_3)`

`color(white)(rArr"area ")=Rxx1/2(b_1+b_2+b_3)`

`color(white)(rArr"area ")=RS`