GMAT Data Sufficiency: Geometry

Q-51 Practice / Data Sufficiency / Geometry / Right Triangles / Q9
The gist of what is tested is the ability to determine whether the data given is sufficient to find a unique set of values for the sides of a triangle. If more than one triangle is possible, then the data will not be sufficient.

Directions

• DS
• 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 July or the meaning of the word counterclockwise), you must indicate whether:

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Question

a, b, and c are sides of a right triangle. What is the area of the triangle?
Statement 1: a = 4.
Statement 2: a + b + c = 12.
Choice C. The two statements together are sufficient to answer the question.

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Detailed Solution

When is the data sufficient?

If we are able to get a UNIQUE value for the area of the triangle from the information given in the statements, the data is sufficient.

What should you watch out for?

Check to see whether more than one set of values are possible with the given data. For e.g., more than one right triangle can be described for a given hypotenuse of 10 units.

Remember that sides of a triangle need not necessarily be integers. Particularly, one might be tempted to think that only triangles whose sides are Pythagorean triplets are right triangles. All Pythagorean triplets are right triangles. But all right triangles need not be Pythagorean triplets. For instance 1, 1, $$sqrt 2 \\$ will form sides of a right triangle, where one of the sides is not an integer. Statement 1 : a = 4 Evaluate statement 1 ALONE It is not possible to find the area of a right triangle with information about just one of its sides and no other information about the triangle. Statement 1 alone is NOT sufficient to answer the question. Because statement 1 is NOT sufficient, we can eliminate choices A and D. Answer choices narrow down to B, C or E. Statement 2 : a + b + c = 12 Evaluate statement 2 ALONE The temptation is to conclude that the sides are 3, 4, and 5. These values will form sides of a right triangle and the sum of a, b, and c is 12. But before concluding that we have a definite answer, let us check whether any right triangle other than 3, 4, and 5 is possible. For instance, it is possible to have a right isosceles triangle whose perimeter is 12. The area of that triangle will be different from that of the triangle with sides 3, 4, and 5. We are not getting a UNIQUE value for the area of the triangle from the information in statement 2. Statement 2 alone is NOT sufficient to answer the question. Because statement 2 is NOT sufficient, we can eliminate choice B as well. Answer choices narrow down to C or E. Combine the statements : a = 4 and a + b + c = 12 If we get a unique set of a, b, and c from these two statements, the data is sufficient In a right triangle, the longest side is the hypotenuse. Side that measures 4 is therefore, not the hypotenuse as 4 is the average of the sum of the 3 sides. The hypotenuse has to necessarily be greater than the average of the 3 sides. So, either b or c is the hypotenuse. Let us assume c to be the hypotenuse. So, 4 and b are the two perpendicular sides of the right triangle. 4 + b + c = 12. So, b + c = 8. Therefore, b = 8 - c Applying Pythagoras theorem, c2 = 42 +$8 - c)2

Solving for c, c2 = 16 + 64 - 16c + c2

16c = 80. So, c = 5.

If c = 5, b = 8 - c = 3

Using the two statements, we could a unique set of values for a, b, and c. Hence, we will be able to find the area of the triangle.

The two statements TOGETHER are sufficient to answer the question.

Choice C is the correct answer.