Sometimes, the best choice from among two options is neither of them!
Q1. A, B, P, and Q are four positive numbers. Does AQ = BP?
Statement #2: In the
You may want to wrestle with this a bit before reading on.
Big geometry idea #1: every point in the
That one fact has several enormous implications. Notice, the distance from the point (x,y) to the origin is the hypotenuse, so we could find that distance with the Pythagorean Theorem.
This should look familiar. That’s exactly the form we have for points
is entirely equivalent to the statement that
Well, how does this help us answer the question: Does
Now, forget about Statement #1 and focus exclusively on Statement #2. We are told the line through
Big geometry idea #2: Two triangles of the same shape and same angles but different sizes are called similar. Similar triangles have proportional sides!! That is one of the deepest and most powerful ideas in all of geometry, and it has staggering implications for problem-solving on the QUANT.
Because the two right triangles are obvious similar, we can set up a proportion among the four sides of interest:
AQ = PB
Voila! It turns out: the prompt question is a proportionality question in disguise. (will you remember that trick on the QUANT?) Statement #2 allows us to answer the prompt with a definitive “Yes!”, so it is sufficient.
Answer = B