Geometry Problem

[coolhabhi]

ABCD is a square and W is the midpoint of side AB. If the side of the square is 16, then the radius of the circle is


A) 2/ [3 + root (5)]
B) 5/ [3 + root (5)]
C) 10/ [3 + root (5)]
D) 15/ [3 + root (5)]
E) 32/ [3 + root (5)]

[abhijit@prepitt]

I wish you had provided an OA. However here goes my attempt. Solving this from scratch will be long so following a couple of methods.

My first instinct is (E). Because the radius most likely is a factor of a multiple of 8. so 32/X looks the part. This is purely a guess.

Now some logic:
From the diagram, the diameter of the circle must lie between 8 & 16 ==> radius of the circle must lie between 4 and 8.

Now lets look at each choice. Assume sqrt(5) ~ 2 or 2.2 (to get ranges)

A) less than 1 [NO]
B) <1 [NO]
C) between 1 and 2 [NO]
D) between 2 and 3 [NO]
E) Already choice by process of elimination . But quick calculation also puts value of E between 6 and 7. This is the only choice within the range of 4 and 8 we expected radius to be.

Hence (E)

[Brent@GMATPrepNow]

ABCD is a square and W is the midpoint of side AB. If the side of the square is 16, then the radius of the circle is


A) 2/ [3 + root (5)]
B) 5/ [3 + root (5)]
C) 10/ [3 + root (5)]
D) 15/ [3 + root (5)]
E) 32/ [3 + root (5)]

On the GMAT, all diagrams in Problem Solving questions are assumed to be in scale, unless it is stated otherwise.
Since the answer choices are quite spread apart, we can ESTIMATE.

Side BC has length 16.
So, the height of the circle is approximately 12, which means the diameter of the circle is about 12.
This means the radius is about 6

When we check the answer choices, we'll use the fact that √5 ≈ 2.2
So 3 + √5 ≈ 5.2

A) 2/5.2 ≈ 0.4 ELIMINATE
B) 5/5.2 ≈ 1 ELIMINATE
C) 10/5.2 ≈ 2 ELIMINATE
D) 15/5.2 ≈ 3 ELIMINATE
E) 32/5.2 ≈ 6 BINGO!!

Answer: E