question_answer

The perimeter of a right triangle is 60 cm. Its hypotenuse is 25 cm. Find the area of the triangle.

Answer:

Given, the perimeter of right triangle \[=60\text{ }cm\]

and hypotenuse \[=25\text{ }cm\]

\[\therefore AB+BC+CA=60\text{ }cm\]

\[AB+BC+25=60\]

\[\therefore AB+BC=35\]                                                            ?(i)

Now, by pythagoras theorem,

\[{{(AC)}^{2}}={{(AB)}^{2}}+{{(BC)}^{2}}\]

\[{{(25)}^{2}}={{(AB)}^{2}}+{{(BC)}^{2}}\]

\[\therefore A{{B}^{2}}+B{{C}^{2}}=625\]                                                   ?(ii)

we, know that, \[{{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\]

then,               \[{{(AB+BC)}^{2}}\text{=(}AB{{)}^{2}}+{{(BC)}^{2}}+2AB.BC\]

\[{{(35)}^{2}}=625+2AB.BC\]

\[\therefore 2AB.BC=1225-625\]

\[2AB.BC=600\]

\[\therefore AB.BC=300\]

\[\therefore \]      Area of \[\Delta \text{ }ABC=\frac{1}{2}~\times AB\times BC\]

\[=\frac{1}{2}\times 300=150\,\,c{{m}^{2}}\]