Question

In a right angled isosceles triangle $\triangle ABC$, right angle at $C$, if side $a = 2$, then sides $b$ and $c$ are respectively

• A. $2\sqrt{2},\, 2$
• B. $\sqrt{2},\, 2$
• C. $2,\, \sqrt{2}$
• D. $2,\, 2\sqrt{2}$

Hint:

In a right isosceles triangle: \[ \text{Sides ratio} = 1 : 1 : \sqrt{2} \] So if one leg is $x$, hypotenuse is always $x\sqrt{2}$.

Answer 1

The Correct Option is D

Step 1: Understanding triangle properties
Since the triangle is right-angled at $C$, side $c$ is the hypotenuse. In a right-angled isosceles triangle, the two perpendicular sides are equal: \[ a = b \] Given: \[ a = 2 \Rightarrow b = 2 \]

Step 2: Applying Pythagoras theorem
For a right-angled triangle: \[ c^2 = a^2 + b^2 \] Substituting values: \[ c^2 = 2^2 + 2^2 = 4 + 4 = 8 \] \[ c = \sqrt{8} = 2\sqrt{2} \]

Step 3: Final answer arrangement
The question asks for $(b, c)$ respectively: \[ b = 2, \quad c = 2\sqrt{2} \] Thus, option (D) is correct.