Question

The area of a right angled triangle is 40 sq.cm and its perimeter is 40 cm.The length of its hypotenuse is

Answer 1

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Answer 2

Given:

Area of a right-angled triangle = 40 sq. cm

Perimeter of the right-angled triangle = 40 cm

To Find:

The length of the hypotenuse of the given right-angled triangle.

Solution:

Let the length of the base of the triangle = b

and the length of the altitude = a

and the length of the hypotenuse = h.

Then since Area of right-angled triangle = $\frac{1}{2} \times base \times altitude$

                                                                   $= \frac{1}{2} \times b \times a = \frac{ab}{2}$

But, given area of triangle = 40

Therefore, comparing both, we get:

$\frac{ab}{2} = 40\\\\\Roghtarrow ab = 2\times40 = 80\\\\\Rightarrow ab = 80$

Since, Perimeter of triangle = Base + Hypotenuse + Altitude

                                                = b + h + a

But, given perimeter of triangle = 40

Therefore, comparing both the values, we get:

$b + h + a = 40\\\\\Rightarrow a + b = 40 - h$

Now, by Pythagoras Theorem,

$(Hypotenuse)^2 = (Altitude)^2 + (Base)^2$

$\Rightarrow h^2 = a^2 + b^2\\\Rightarrow h^2 = (a+b)^2 - 2ab$      $[ \because (a+b)^2 = a^2 + b^2 +2ab \ ]$

Now, putting the values from the obtained equations, we get:

$h^2 = (40 -h)^2 - 2\times80\\\\\Rightarrow h^2 = 40^2 + h^2 - 2 \times 40\times h - 160 \ \ [As (a-b)^2 = a^2 +b^2 -2ab]\\\\\Rightarrow h^2 - h^2 + 80h = 1600 -160\\\\\Rightarrow 80h = 1440\\\\\Rightarrow h = \frac{1440}{80} = 18$

Therefore, we get the length of the hypotenuse = 18 cm.