The given question is a GMAT 700 level data sufficiency question. We have 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 is not sufficient.
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: 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.
Q1. 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.
Q2. What should you watch out for?
1. 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.
2. 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, √2 will form sides of a right triangle, where one of the sides is not an integer.
Statement 1: a = 4
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.
We are not able to find the area using statement 1.
Hence, statement 1 is not sufficient.
Eliminate answer option A and D.
Statement 2: a + b + c = 12
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 able to find a UNIQUE area using statement 2.
Hence, statement 2 is not sufficient.
Eliminate answer option B.
Statements: a = 4 & a + b + c = 12
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 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 get a unique set of values for a, b, and c.
Hence, we will be able to find the area of the triangle.
Statements together are sufficient to find a UNIQUE value as the area of the right triangle.
Eliminate answer option E.
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