What is the length of CD in the quadrilateral ABCD shown below?
I) The length of AB is 10 cm and BC is 16 cm.
II) BC is perpendicular to BD, and AD is perpendicular to AB.
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In geometry DS questions, combine lengths with angle conditions to form right triangles and apply Pythagoras.
Statement I alone is sufficient to answer the question
Statement II alone is sufficient to answer the question.
Statement I and II together are sufficient to answer the question but neither statement alone is sufficient.
Statement I and II together are not sufficient to answer the question and additional data is required.
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The Correct Option isC
Solution and Explanation
Concept:
To determine a length in geometry, we need sufficient measurements and angle/relationship constraints. Right angles and given lengths help form triangles where Pythagoras can be applied.
Step 1: Analyze Statement I alone.
Given:
\[
AB = 10 { cm}, \quad BC = 16 { cm}
\]
No angle or perpendicularity information is given. The shape is not fixed. Statement I alone is NOT sufficient. Step 2: Analyze Statement II alone.
Given:
\[
BC \perp BD, \quad AD \perp AB
\]
Only angular relationships are given, no lengths. Statement II alone is NOT sufficient. Step 3: Combine both statements.
Now we have:
\(AB = 10\), \(BC = 16\)
\(AD \perp AB\) $\Rightarrow$ right angle at A
\(BC \perp BD\) $\Rightarrow$ right angle at B in triangle BCD
Angle at B between AB and BD is \(45^\circ\) (from figure)
Using triangle ABD:
\[
\angle ABD = 45^\circ,\quad AB = 10
\]
So,
\[
BD = \frac{AB}{\cos 45^\circ} = \frac{10}{\frac{1}{\sqrt{2}}} = 10\sqrt{2}
\]
Now in triangle BCD (right-angled at B):
\[
CD^2 = BC^2 + BD^2 = 16^2 + (10\sqrt{2})^2 = 256 + 200 = 456
\]
\[
CD = \sqrt{456} = 2\sqrt{114}
\]
A unique value is obtained.
Hence, both statements together are sufficient, but neither alone is sufficient.
