The circle enclosed in the square HIJK above touches only two points

Question

https://gmatclub.com/forum/download/file.php?id=43460 The circle enclosed in the square HIJK above touches only two points on the perimeter of the square, marked X and Y, and both are equidistant from point K. What is the radius of the circle? (1) The sum of segment XK and segment YK is 16. (2) The length of the radius is two-fifths the length of any side of the square. image010.jpg

Nirenjan · Answer

If the image is in scale answer is D

Sent from my iPhone using  GMAT Club Forum mobile app

Varthini · Answer

The answer is E , bcos

Statement 1, => xk=8, yk=8 since x and y are equidistant from k, but it cannot be used to find the radius. - insufficient

Statement 2 => radius = (2/5)(side) but length of 1 side is not given - insufficient

Both Together, cannot give the length of the side or the radius.

Hence E

karnaidu · Answer

From the given information , we can easily conclude that the given tangents KY and KX form a square if we just connect the point of contacts from circle's center .
So if we know the length of the KY or KX , we can find the radius .   .

From statement 1 , we can find the radius => Sufficient 
Statement 2 => not sufficient

Answer is A

PKN · Answer

Question stem:- Radius=OX=OY=?

St1:-  The sum of segment XK and segment YK is 16. 
XK+YK=16
Or, XK=YK=8 (Since the circle touches the square at its adjacent sides , hence XK=YK)
Notice that OY=XK, so OY=8 (Please refer the attachment)
Similarly, OX=YK, so, OX=8
Sufficient.

St2:-  The length of the radius is two-fifths the length of any side of the square. 
Since measure of side of square is not known, hence radius can't be determined.
Insufficient.

Ans. (A)

rajudantuluri · Answer

Nowhere does it say that X is at 90 degrees from O. For all we know, couldn't the center be slightly towards the left from X?

PKN · Answer

Hi  rajudantuluri ,

Given, sides of the square, HK and JK , touch the circle at the points X and Y respectively, implies that H-X-K and K-Y-J are the tangents.
 DEFINITION OF Tangent :
A line passing a circle and touching it at just one point. The tangent line is always at the 90 degree angle (perpendicular) to the  radius  of a circle. 
You know radius is the distance from the center to any point on the circle.

Therefore the perpendicular lines XO and YO meet at the center of circle.

Hope it helps.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

The circle enclosed in the square HIJK above touches only two points

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards