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. A GMAT hard math question.
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 9: Is triangle ABC with sides a, b, and c acute 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 the statements independently or together do not provide a DEFINITE YES or DEFINITE NO, the data is NOT sufficient.
Q3. What do we know from the question stem?
The question stem states that a, b, and c are the measures of the sides of a triangle.
Key property of acute angled triangles
If a, b, and c are the measures of the sides of a triangle, and if 'a' is the longest side of the triangle, then
- the triangle is acute angled if a2 < b2 + c2
- right angled if a2 = b2 + c2 and
- obtuse angled if a2 > b2 + c2
The statement provides us with one valuable information: we can form a triangle with sides a2, b2, c2.
For any triangle we know that sum of two sides is greater than the third side.
So, we can infer that a2 < b2 + c2.
The inequality above is the condition to be met if the triangle with sides a, b, and c were to be an acute triangle.
We are able to answer the question with a DEFNITE answer using Statement 1.
Hence, statement 1 alone is sufficient.
Choices narrow down to A or D.
For an equilateral triangle, medians to the sides of the triangle are the corresponding altitudes. i.e., the median and altitude of all 3 sides are coincident lines.
For an isosceles triangle, the median to the side whose measure is different is the altitude to that side. i.e., only one median is the same as the altitude.
From statement 2, we can infer that the triangle is either equilateral or isosceles.
An equilateral triangle is definitely an acute angled triangle. However, an isosceles triangle need not be an acute angled 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.
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