In a DS question on Geometry, do not make any assumptions and try to see if you can obtain a unique diagram. If you overlook any of these two rules, you may end up falling for the trap answer.
For example, in this question, although the triangles look like Right triangles, I’d only decide after reading the first sentence. Had the first sentence not mentioned that the triangle is a right triangle, we cannot assume that they are right angled triangles, just because they look like.
‘Looks-like’ has been the undoing on Geometry questions for a lot of students.
Also, I know that angles C and M are right angles in the respective triangles, only after I read the second statement in the question. Till this stage, I will not assume that C and M are right angles.
From the question statements, we know that,
Area of triangle ABC = 4 * Area of triangle KLM. Now, does this mean that triangles ABC and KLM are similar?? If you said yes, you have fallen for the trap. The answer is “Not necessarily”.
Hypotenuses of the respective right triangles are AB and KL, hence the right angles are at C and M respectively.
When you have 2 triangles with one set of angles equal, you should think of proving the triangles similar by finding another set of equal angles. Once this is done, you will be able to use the properties of similar triangles to solve the question.Statement I alone says that angles ABC and KLM are equal. We now have the second set of angles that we were looking for. We can now say that the triangles ABC and KLM are similar. Therefore,
\(\frac{Area of triangle ABC }{ Area of triangle KLM}\) = \(\frac{{AB^2}}{{KL}}\). Since we know the ratio of the areas of the triangles and also KL, we can find out the length of AB from this data.
Statement I alone is sufficient. Possible answer options are A or D. Answer options B, C and E can be eliminated.
Statement II alone says that LM is 6 inches. This can only help us determine the other side of the triangle i.e. KM = 8 inches. We may also be able to establish the area of triangle KLM and hence the area of triangle ABC.
However, this is all we can do. We do not have any other link between the two triangles in terms of angles so that we could prove them similar. Hence, we will not be able to find out AB.
Statement II alone is insufficient. Answer option D can be eliminated.
The correct answer option is A.
Note that the question statement mentions '4 times greater than'; technically, this should mean ' Area of triangle ABC = 5* Area of triangle KLM'.
However, when we tie in this information with the other data like the lengths of the hypotenuse and the areas, we see that 'Area of triangle ABC = 4* Area of triangle KLM' makes more sense since we get Area of triangle KLM = 24 sq.units and Area of triangle ABC = 96 sq.units when we solve using statement I alone.
Hope that helps!
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Crackverbal Prep Team
www.crackverbal.com