In `DeltaABC`, if `sin^(2)A+sin^(2)B=sin^(2)C and l(AB)=10,` then the maximum value of the area of `DeltaABC` is

Options

A. 50
B. `10sqrt(2)`
C. 25
D. `25sqrt(2)`

Answer

C

Explanation

Given, `sin^(2)A+sin^(2)B=sin^(2)C`
`impliesa^(2)+b^(2)=c^(2)` (By sine rule)
which is result of Pythagoras theorem.
`thereforeDeltaABC` is right angled triangle and right angled at C.
Now, area of `(DeltaACB)=(1)/(2)ab` . . . (i)

[`because`Area=`(1)/(2)`Base`xx`height]
From sin rule's, we have
`(a)/(sinA)=(b)/(sinB)=(c)/(sinC)`
`implies(a)/(sinA)=(b)/(sinB)=(10)/(1)`
`impliesa=10sinA`
`b=10sinB`
On substituting a=10 sinA and b=10sinB in Eq. (i),
we get
Area of `DeltaACB=(1)/(2)(10sinA)(10sinB)`
=50 sin A sin B . . . (ii)
But maximum value of sin A sin B`=(1)/(2)` . . . (iii)
`therefore`Maximum value of area of `DeltaACB=50xx(1)/(2)=25` [from eqs. (ii) and (iii)]