# If A(-2,5) , B(1,-3) and C(a,b) form an isosceles triangle then find the value of 6a-16b+19.

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If A(-2,5) , B(1,-3) and C(a,b) form an isosceles triangle then find the value of 6a-16b+19.

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Solution:

We have given A(-2,5) , B(1,-3) and C(a,b) form an isosceles triangle then the value of 6a-16b+19 = ?

We know the distance formula between two points

Now

AB = sqrt[(-3 - (5))² + ((1) - (-2))²]

AB = √73

BC = sqrt[(b - (-3))² + ((a) - 1)²] = sqrt[(b + 3))² + (a - 1)²]

AC = sqrt[(b - 5))² + (a + 2)²]

Since triangle is isosceles, and we know isosceles triangle have two sides are equal, so either AB=BC or AB=AC or BC=AC

Let AB = BC

√73 = sqrt[(b + 3))² + (a - 1)²]

73 = (b + 3))² + (a - 1)²  ----------(i)

and AB = AC

√73 = sqrt[(b - 5))² + (a + 2)²]

73 = (b - 5))² + (a + 2)²  -------------(ii)

From equation (i) and (ii), we get

=> (b + 3))² + (a - 1)²  = (b - 5))² + (a + 2)²

Expanding the equation we get

6a - 16b + 19 = 0   Ans.