Area of a right-angled triangle is 6 cm^2 and its perimeter is 12 cm. Then length of its hypotenuse, is

Q: Area of a right-angled triangle is $6 cm^2$ and its perimeter is 12 cm. Then length of its hypotenuse, is

(a) 5 cm Let the lengths of the perpendicular sides of the right-angled triangle be a cm and b cm. Then, its hypotenuse is of length $\sqrt{a^2+b^2} cm$. It is given that the perimeter of the triangle is 12 cm and area is $6 cm^2$. $\therefore \quad \frac{1}{2} a b=6 \Rightarrow a b=12$$\quad$$\ldots$ (i) and, $a+b+\sqrt{a^2+b^2}=12$ $\Rightarrow a+b+\sqrt{(a+b)^2-2 a b}=12$ $\Rightarrow x+\sqrt{x^2-2 \times 12}=12$, where $x=a+b$ $\Rightarrow \sqrt{x^2-24}=(12-x)$ $\Rightarrow x^2-24=(12-x)^2 \Rightarrow x^2-24=144-24 x+x^2 \Rightarrow 24 x=168 \Rightarrow x=7 \Rightarrow a+b=7$$\quad$...(ii) $\therefore \quad Hypotenuse =\sqrt{a^2+b^2}=\sqrt{(a+b)^2-2 a b}=\sqrt{7^2-2 \times 12}=\sqrt{25}=5 cm$.

M.C.Q

CBSE - English Medium

Area of a right-angled triangle is $6 cm^2$ and its perimeter is 12 cm. Then length of its hypotenuse, is

A
5 cm
B
6 cm
C
7 cm
D
8 cm

STD 9
Maths
Herons Formula
RD Sharma

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