Question

a circle with Centre P is inscribed in a triangle ABC side a b side BC and side AC touch the circle at points L,m and n respectively the radius of the circle is r
prove that area of triangle ABC =1/2(AB+BC+AC)×r​

Answer 1

Answer:

Proved

Step-by-step explanation:

a circle with Centre P is inscribed in a triangle ABC side a b side BC and side AC touch the circle at points L,m and n respectively the radius of the circle is r

prove that area of triangle ABC =1/2(AB+BC+AC)×r​

Let say center point = O

if we draw line from points A , B & C  at point O

we can Divide ΔABC into three triangle

ΔAOB  , ΔBOC & ΔCOA

Area of ΔAOB =  (1/2) * AB * OL  ( Base * Perpendicular)

OL = Radius = r

Area of ΔAOB =  (1/2) * AB * r

SImilarly

Area of ΔBOC =   (1/2) * BC * r

Area of ΔCOA =   (1/2) * AC * r

Area of ΔABC = Area of ΔAOB + Area of ΔBOC  + Area of ΔCOA

=> Area of ΔABC =  (1/2) * AB * r +  (1/2) * BC * r  +  (1/2) * AC * r

=> Area of ΔABC =  (1/2) * (AB + BC + AC) * r

QED