GMATPrep Geometry; number theory

Question

Geometry question that I am stuck on...

Seems quite easy, However I just can't put my finger on the theory.


OA will be provided later...

jaynayak · Answer

s = 1

Distance between OP and OQ is same. (radius)

OP = 2 (use distance between two points formula.One point is the origin)

Hence PQ = 2(2)^1/2

OPQ is isosceles right triangle.


Now we pick numbers from answer choices. Find t and check whether PQ = 2(2)^1/2 for that set.

rdw28 · Answer

Correct! Thanks alot Jaynayak

jamesrwright3 · Answer

I am slightly confused
What is the distance between points theory you allude to?

v1rok · Answer

Here is my solution (without plugging numbers):

Let's drop vertical lines from P and Q onto x-axis, and let's denote corresponding intersection ponts as P' and Q'.

We know that the length of OP'=sqrt(3). We need to find out the length of OQ' (or s).

First, since radius equals to 2, then PO=2. Now note that triangle PP':PO as 1:2. Hence it is a 30-60-90 triangle. In fact, <POP'=30"

Now, from the fact that sum of all angles in a triangle =180, it can be shown that <OQQ'=30" and <QOQ' = 60", So, OQQ' is also 30-60-90 triangle.

Hence s=OQ'=1.

Zooroopa · Answer

Another tedious solution, but needing no extra knowledge!! Use Pythagoras's theorem about rt angled traingles.

s^^2 + t^^2 = 4 ....(i)
s^^2 + t^^2 + 3 + 1 = (s+√3)^^2 + (t-1)^^2 ...(ii)

t = √3s

So s^^2 = 1 ---> s = 1 as 's' is in the positive quadrant!!

GMATPrep Geometry; number theory

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