This is an easy Data Sufficiency question on the properties of the two special Right Angled Triangles – the 30-60-90 Right Angled Triangle and the 45-45-90 Right Angled Triangle.
In a 30-60- 90 Right Angled Triangle, when the side opposite to the 30 degree angle is ‘x’, the side opposite to the 60 degree angle will be x√3 and the side opposite to the 90 degree angle i.e. the hypotenuse will be 2x. So, if we know the angles and one of the sides, we can find out the other two sides. A general 30-60-90 triangle is depicted below:
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In a 45-45- 90 Right Angled Triangle, if the sides opposite to the 45 degree angles are equal to x, the side opposite to the 90 degree angle i.e. the hypotenuse will be x√2. In this case also, if we know the angles and one of the sides, we can find out the other two sides. A general 45-45-90 triangle is depicted below:
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Now, in this question, we know that triangle ACB is a 30-60- 90 right angled triangle with angle CAB = 90 degrees (note that the question says CB is the common hypotenuse; the hypotenuse is always opposite to the right angle) and angle ACB = 60 degrees. Therefore, angle ABC = 30 degrees.
Hence, if AC = x, AB = x√3 and BC = 2x.
Also, triangle BCD is a 45-45-90 right angled triangle (isosceles right angled triangle) with angle CDB = 90 degrees (again, the same reasoning that BC is the hypotenuse) and angle CBD = angle BCD = 45 degrees.
Hence, CD = DB = x and BC = 2x.
From the above analysis of the question, we can understand that if any one of the sides of either of the triangles is given, we can always find out all the other sides. If we are able to find the exact lengths of the sides, we will always be able to find the difference between AB and DC.
From statement I alone, BC = 12. This means, 2x = 12. DC and AB can be found out uniquely and hence a unique difference can be established. Sufficient.
Possible answer options are A or D. Answer options B, C and E are eliminated.
From statement II alone, CA is 6. This means x = 6 from which AB and DC can be found out and so can be the difference between the two. Sufficient.
Answer option A can be eliminated, the right answer has to be D.
In such questions, spend time on analyzing and simplifying the question data and the question stem. Bring the question down to a stage where you know exactly what to look for in the statements. You will then be able to make the best use of the data given in the statements.
Hope that helps!
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