Question

In a right angled isosceles triangle \(\Delta ABC\), 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 \(45^\circ-45^\circ-90^\circ\) triangle, \[ \text{Hypotenuse} = (\text{Leg})\sqrt2. \] If each equal side is \(a\), then the hypotenuse is always \(a\sqrt2\).

Answer 1

The Correct Option is D

Concept: In a triangle, the sides opposite vertices \(A\), \(B\), and \(C\) are denoted by \(a\), \(b\), and \(c\) respectively. Since the triangle is right-angled at \(C\), \[ \angle C = 90^\circ, \] the side opposite the right angle, namely \(c\), is the hypotenuse. Further, because the triangle is an isosceles right-angled triangle, the two legs containing the right angle are equal in length. Therefore, \[ a=b. \] The hypotenuse can then be obtained using the Pythagorean theorem.

Step 1: Use the property of an isosceles right triangle. Since the triangle is isosceles and right-angled at \(C\), \[ a=b. \] Given, \[ a=2. \] Therefore, \[ b=2. \]

Step 2: Apply the Pythagorean theorem. For a right triangle, \[ c^2=a^2+b^2. \] Substituting \(a=2\) and \(b=2\), \[ c^2=2^2+2^2. \] \[ c^2=4+4=8. \]

Step 3: Calculate the hypotenuse. Taking square roots on both sides, \[ c=\sqrt8 =\sqrt{4\times2} =2\sqrt2. \]

Step 4: Write the values of \(b\) and \(c\). We obtained \[ b=2 \] and \[ c=2\sqrt2. \] Conclusion: Hence, the values of \(b\) and \(c\) respectively are \[ \boxed{2,\;2\sqrt2}. \] Therefore, the correct option is \[ \boxed{(D)}. \]