Question : If the perimeter of an isosceles right triangle is $8(\sqrt{2}+1)$ cm, then the length of the hypotenuse of the triangle is:
Option 1: 8 cm
Option 2: 12 cm
Option 3: 10 cm
Option 4: 24 cm
Correct Answer: 8 cm
Solution : Given: The perimeter of an isosceles right triangle is $8(\sqrt{2}+1)$ cm.
The total of a triangle's sides is its perimeter.
Let the isosceles right triangle have equal sides of $x$ cm, then its hypotenuse is $\sqrt2x$ cm.
According to the question,
$x+x+\sqrt 2x=8(\sqrt{2}+1)$
⇒ $2x+\sqrt 2x=8(\sqrt{2}+1)$
⇒ $\sqrt2x(\sqrt 2+1)=8(\sqrt{2}+1)$
⇒ $x=\frac{8}{\sqrt2}$
⇒ $x=4\sqrt2$ cm
The length of the hypotenuse of the triangle = $\sqrt2\times 4\sqrt2=8$ cm.
Hence, the correct answer is 8 cm.
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