In a `DeltaABC, b = 12` units, c = 5 units and `Delta = 30`sq. units. If d is the distance between vertex A and incentre of the triangle then the value of `d^(2)` is _____

Answer

8

Explanation

`AI = r "cosec" (A)/(2) = (Delta)/(s) "cosec" (A)/(2)`
Also, `Delta = (1)/(2) bc sin A`
`rArr 30 = (1)/(2) xx 12 xx 5 xx sin A`
`rArr sin A = 1`
`rArr A = (pi)/(2)`
`:. A^(2) = b^(2) + c^(2)`
`rArr a = 13`
`:. S = (a +b +c)/(2) = 15`
Hence, `AI = 2 sqrt2` units