buoyant wrote:The length of the median BD in triangle ABC is 12 centimeters, what is the length of side AC?
(1) ABC is an isosceles triangle
(2) AC^2 = AB^2 + BC^2
[spoiler]OA: B[/spoiler]
Well I messed around with this.
First I took care of Statement 1. For one thing it does not say which of the sides are equal, or much else. So we can't figure out the relationship between the length of the median and that of the side AC.
Insufficient
Statement 2 is interesting. From the equation we can tell that ABC is a right triangle with AC as the hypotenuse, because the squares of the lengths of the other two sides add up to the square of AC. Then I played with some different triangles. If ABC is a 45-90-45 triangle, the median extends down and creates two new 45-45-90 triangles. The two equal sides of each of these triangles are the median and half of AC. So if the length of the median is half of AC, then AC = 2 * 12 = 24.
Alternatively if ABC is a 30-90-60 triangle, the median will form an equilateral, 60-60-60, triangle and a 30-120-30 triangle. Hmm. So once again the median is half of AC and so AC is 24.
So I am guessing that with some more math we will discover that a median extending from the right angle of a right triangle always forms two isosceles triangles with one of the two equal sides of each triangle being the median and the other being half the hypotenuse.
Yup, I looked it up, and by inscribing the right triangle in a circle we can show that the median is always half the hypotenuse.
So Statement 2 is sufficient and the answer is
B.
Having said that, I somehow doubt having to prove a geometric rule in this way would be necessary to answer a two minute GMAT question, and further, I don't believe this rule is part of the body of knowledge one is expected to know in order to have the basis for answering GMAT questions.
In other words, while that rule might occasionally be useful in answering a GMAT question, one does not need to know it or derive it to solve GMAT problems.
Maybe someone else will have a different opinion, but that's my take. In a way it's a cool question, and maybe something to learn from, but that's it.
