A GMAT Data Sufficiency practice question in Geometry. Concept: properties of sides of Obtuse & Acute angled triangles and properties relating to location of the center of the circumscribing circle of a triangle.
This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -
All numbers used are real numbers.
A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2)
Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight
You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.
All figures lie in a plane unless otherwise indicated.
In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.
Question 8: Is triangle ABC obtuse angled?
Q1. What kind of an answer will the question fetch?
An "IS" question will fetch a definite "Yes" or a "No" as an answer.
Q2. When is the data sufficient?
The data is sufficient if we are able to get a DEFINITE YES or a DEFINITE NO from the information given in the statements.
If we get an answer that the triangle may be obtuse in some cases and not obtuse in other cases, the data is not sufficient.
Key property about sides of an obtuse triangle
If 'c' is the longest side, for an obtuse triangle, c2 > a2 + b2.
Conversely, if c2 < a2 + b2, the triangle is acute.
This statement does not tell us whether 'c' is the longest side in the triangle.
Note: If 'c' were the longest side we could have deduced that the given triangle is an acute angled triangle.
We are unable to answer the question with a DEFNITE answer using Statement 1.
Hence, statement 1 alone is NOT sufficient.
Choices narrow down to B, C or E.
Property of location of the center of a circumscribing circle
1. For an acute angled triangle, the center of the circle circumscribing the triangle lies inside the triangle.
2. For a right triangle, the center of the circle circumscribing the triangle lies at the mid point of the hypotenuse.
3. For an obtuse triangle, the center of the circle circumscribing the triangle lies outside the triangle.
From statement 2, we can deduce that the triangle is not an acute angled triangle. It may be a right triangle or it could be an obtuse triangle.
We are unable to answer the question with a DEFNITE YES/NO using Statement 2.
Hence, statement 2 alone is not sufficient.
Eliminate answer option B.
The center of the circle circumscribing the triangle does not lie inside the triangle.
From statement 2 we know that the triangle is not acute. Without knowing whether 'c' is the longest side, we will not be able to conclude whether the triangle is obtuse.
Eliminate choice C.
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