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A circle with centre P is inscribed in the A ABC. Side AB, side BCand side AC touch the circle at points L, M and N respectively. Radiusof the circle is r.Prove that:ACA ABC) (AB + BC + AC)xr.n |
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Answer» 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 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 |
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