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Q.
Area of a right-angled triangle is $6 \,cm ^2$ and its perimeter is $12 cm$. Find its hypotenuse. (in cm)
Mensuration
Solution:
Let the sides of the triangle (in $cm$ ) be $a, b$, and $c$, where $a
$\therefore c$ is the hypotenuse and $a$ and $b$ are the perpendicular sides.
Given, $a+b+c=12$(1)
$\frac{1}{2}(a)(b)=6$(2)
From Eq. (1) $\Rightarrow a+b=12-c$
Squaring both sides,
$a^2+b^2+2 a b=12^2-24 c+c^2$
$=a^2+b^2+4\left(\frac{1}{2} a b\right)=12^2-24 c+c^2$
$\Rightarrow c^2+4(6)=12^2-24 c+c^2$
$c=5$ .