Question : ABC is an isosceles triangle having $\angle$ C = 90$^\circ$, if D is any point on AB, then AD2 + BD2 is equal to:
Option 1: CD2
Option 2: 2CD2
Option 3: 3CD2
Option 4: 4CD2
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Correct Answer: 2CD 2
Solution : Given that ABC is an isosceles triangle with $\angle$ C = 90$^\circ$. Construct a circle of radius $r$ circumscribing triangle ABC with AB as diameter as $\angle$C will be an angle in a semicircle. Since D can be any point, take it as the center of the circle. By construction, AD = BD = CD = $r$ So, AD 2 + BD 2 = 2$r^2$ or, AD 2 + BD 2 = 2CD 2 Hence, the correct answer is 2CD 2 .
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