Question

Find the length of AB if $\angle YBC = \angle CAX = \angle YOX = 90^\circ$.
[I.] Radius of the arc is given.
[II.] \( OA = 5 \)

• A. if the question can be answered with the help of any one statement alone but not by the other statement.
• B. if the question can be answered with the help of either of the statements taken individually.
• C. if the question can be answered with the help of both statements together.
• D. if the question cannot be answered even with the help of both statements together.

Hint:

If multiple right angles are given but critical point positions are not determined uniquely, the length cannot be calculated.

Answer 1

The Correct Option is D

We are asked to find the length of \( AB \), where points lie on a geometric figure involving a quarter circle (with center \( O \)), and three right angles are given at \( \angle YBC, \angle CAX, \angle YOX = 90^\circ \). Statement I: Radius of the arc is given.
This gives us the total length \( OX \), but it does not tell us anything directly about where point \( B \) lies, or how triangle \( ABO \) is structured. So, not sufficient. Statement II: \( OA = 5 \)
This gives one side, but again, without knowing the radius or coordinates of point \( B \), it is not enough to determine length \( AB \). Still not sufficient. Combining both: Even with both, we know OA = 5 and radius = 5 (since arc is centered at O and touches X), but we do not have a complete triangle to determine AB without further positional information about B and C. Hence, we cannot find the exact length of AB even with both statements.