Question : The perimeter of a certain isosceles right triangle is $10+10\sqrt{2}$ cm. What is the length of the Hypotenuse of the triangle?
Option 1: $5$ cm
Option 2: $10$ cm
Option 3: $5\sqrt{2}$ cm
Option 4: $10\sqrt{2}$ cm
Correct Answer: $10$ cm
Solution :
\(\text{Perimeter} = \text{Sum of all sides}\)
Sides of an isosceles right triangle are $a, a, and $\sqrt{2}a$$
So, \(\text{Perimeter} = a+a+\sqrt{2}a \)
\(=10+10\sqrt{2} \)
⇒ \((2a+\sqrt{2}a)=(10+10\sqrt{2}) \)
⇒ \(\sqrt{2}a(1+\sqrt{2})=10(1+\sqrt{2}) \)
⇒ \(a=\frac{10}{\sqrt{2}} \)
⇒ BC (Hypotenuse) = \(\sqrt{2}a\)
⇒ \(\frac{\sqrt{2} \times 10}{\sqrt{2}}=10\ \mathrm{cm} \)
Hence, the correct answer is $10$ cm.
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