moonwalking wrote:
A triangle has side lengths of a, b, and c centimeters. Does each angle in the triangle measure less than 90 degrees?
(1) The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 \(cm^2\), 4 \(cm^2\), and 6 \(cm^2\), respectively.
(2) c < a + b < c + 2
Target question: Does each angle in the triangle measure less than 90 degrees? Given: A triangle has side lengths of a, b, and c centimeters Statement 1: The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 \(cm^2\), 4 \(cm^2\), and 6 \(cm^2\), respectively. This statement illustrates an important rule concerning Data Sufficiency questions: DO NOT DO MORE WORK THAN IS NECESSARY
Since we have the area for each circle, we COULD calculate the diameter of each circle.
So, after some tedious calculations, we WOULD know the lengths of all 3 sides of the triangle.
IF we know the lengths of all 3 sides of the triangle, we COULD draw the triangle, and we COULD then measure each angle with a protractor.
This means we COULD definitely determine whether each angle is less than 90 degrees.
In other words, we COULD answer the
target question with certainty.
This means statement 1 is SUFFICIENT
Statement 2: c < a + b < c + 2There are several values of a, b and c that satisfy statement 2. Here are two:
Case a: a = 1, b = 1 and c = 1. In this case, we have an EQUILATERAL triangle in which all three angles measure 60 degrees. So, the answer to the target question is
YES, each angle in the triangle measures less than 90 degreesCase b: a = 1.5, b = 2 and c = 2.5.
ASIDE: We know that a 3-4-5 triangle is a RIGHT triangle.
If we make each side HALF as long, we get a 1.5-2-2.5 triangle, which is also a RIGHT triangle.
In this case, we have a RIGHT triangle which means one angle EQUALS 90 degrees. So, the answer to the target question is
NO, it is NOT the case that each angle in the triangle measures less than 90 degreesSince we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A
Cheers,
Brent