In ΔABC if sin² A + sin² B = sin² C and \(l (AB) =10\)  then the maximum value of the area of ΔABC is 

Concept:

In ΔABC, sine rule

\(\rm \dfrac {a}{sin\; A} = \dfrac {b}{sin\; B} = \dfrac {c}{sin\;C} \)

\(\rm A (ΔABC) = \dfrac{1}{2}\times base \;\times height \)

Calculations:

Given, In ΔABC if sin2 A + sin2 B = sin2 C and

length of Side (AB) = 10 ⇒ c = 10

⇒ a2 + b2 = c2

Pythagoreans theorem is verified.

\(\rm \angle C = 90^\circ\)

Hence,  ΔABC is right angled triangle.

\(\rm A (ΔABC) = \dfrac{1}{2}\times base \;\times height \)

\(\rm A (ΔABC) = \dfrac{1}{2}\times a \times b\)....(1)

By sine rule, we have

\(\rm \dfrac {a}{sin\; A} = \dfrac {b}{sin\; B} = \dfrac {c}{sin\;C} \)

⇒\(\rm \dfrac {a}{sin\; A} = \dfrac {b}{sin\; B} = \dfrac {10}{sin\; 90^\circ} \)

⇒\(\rm \dfrac {a}{sin\; A} = \dfrac {b}{sin\; B} = \dfrac {10}{1} \)

⇒\(\rm a = 10\;sin\;A\) and \(\rm b = 10\;sin\;B\)

 Equation (1) becomes, 

\(\rm A (ΔABC) = \dfrac{1}{2}\times 10 \;sin\; A \times 10\;sin\; B \)

⇒\(\rm A (ΔABC) = 50 \times sin\; A \times sin\; B \) ....(2)

Since, \(\rm \angle C = 90^\circ\)and sum of all angles of triangle is 180º

\(\rm \angle (A+B) = 90^\circ\)

⇒Max (\(\rm sin\; A \times sin\; B\)) = \(\rm \dfrac 1 2\)

Equation (2) becomes,

\(\rm A (ΔABC) = 50 \times \dfrac 1 2 \)

\(\rm A (ΔABC) = 25 \)