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

Concept: In a right-angled isosceles triangle, the two sides forming the right angle are equal in length. These two equal sides are called the legs, and the side opposite the right angle is the hypotenuse.

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.