Incircle of radius of a triangle touches the side at . If , then what area of triangle ?
Give your answer in .
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Let the incircle touch AC at E, AB at F. Tangents from vertex to in circle have equal lengths. ∴ AE=AF=X say. FB=BD=6. EC=DC=8. ⟹ AB=X+6. BC=6+8. AC=X+8. So semiperemeter S=X+14. ∴ A r e a 2 = ( X + 1 4 ) ∗ 8 ∗ X ∗ 6 . . . . . H e r o ′ s f o r m u l a . ∴ i n r a d i u s 2 = S 2 a r e a 2 , ⟹ 4 2 = ( X + 1 4 ) 2 ( X + 1 4 ) ∗ 8 ∗ X ∗ 6 S o 1 6 = 4 8 ∗ ( X + 1 4 ) , ⟹ X = 7 . ∴ a r e a = ( 7 + 1 4 ) ∗ 8 ∗ 7 ∗ 6 = 8 4 c m 2 .