A GMAT DS question in Geometry. Concept: This is an excellent question to get clarity on the following properties: Properties of sides of an acute triangle, properties relating to median and altitude of equilateral and isosceles triangles.
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 -
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- EACH statement ALONE is sufficient to answer the question asked.
- Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
All numbers used are real numbers.
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.
- Triangle with sides a2, b2, c2 has an area of 140 sq cms.
- Median AD to side BC is equal to altitude AE to side BC.
Explanatory AnswerVideo explanation will be added soon
What kind of an answer will the question fetch?
The question is an "Is" question. Answer to an "is" questions is either YES or NO.
When is the data sufficient?
The data is sufficient if we are able to get a DEFINTE YES or DEFINITE NO as the answer.
If the statements independently or together do not provide a DEFINITE YES or DEFINITE NO, the data is NOT sufficient.
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 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
Statement 1: Triangle with sides a2, b2, c2 has an area of 140 sq cms.
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.
Statement 1 ALONE is sufficient.
Eliminate choices B, C and E. Choices narrow down to A or D.
Statement 2: Median AD to side BC is equal to altitude AE to side BC.
Equilateral and Isosceles triangle properties
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.
Statement 2 ALONE is NOT sufficient.
Statement 1 is sufficient, while statement 2 is not sufficient.
Eliminate choice D. Choice A is the answer.