A circle is inscribed in a right triangle as shown. If OC + OE = 2.4

Question

Capture.PNG A circle is inscribed in a right triangle as shown. If OC + OE = 2.4, which of the following is closest to the circumference of the circle? (Note: Figure not drawn to scale.) A. 3.2 B. 3.6 C. 6.3 D. 6.8 E. 7.2

GMATinsight · Accepted Answer

As per the figure you need to understand

Property 1: The Line joining the Centre of the circle and the Point of Tangency of another line will always make 90 degrees

Property 2: From any exterior point only two tangents can be drawn on any specific circle and both the Tangents will be Equal

i.e. (According to Property 1) OE will be perpendicular to AB and OD will be perpendicular to AC

Also, OE = OD = Radius of the Circle

If we Draw a line OF then angle C, angle D and Angle F will be Right angles and OD = OF = radius therefore ODCF will be a Square (all angles 90 degrees and adjacent sides are equal)

Now Triangle ODC will be an Isosceles Right angle triangle with OD = DC = radius ( r ) and Hypotenuse OC= r\sqrt{2}

Now, OC + OE = 2.4
i.e. OC + OD = 2.4
i.e.  r+r\sqrt{2} = 2.4 
i.e.  r+1.4r = 2.4  ( \sqrt{2}=1.4  approximately)
i.e.  r=1

i.e. Circumference = 2(pi)r = 2(pi) = 2*3.14 = 6.28

Answer: Option  C

zahraf · Answer

I can not understand this part of your solution
i.e. $r+r\sqrt{2} = 2.4$
i.e. $r+1.4r = 2.4$ ($\sqrt{2}=1.4$ approximately)
i.e. $r=1$

i.e. Circumference = 2(pi)r = 2(pi) = 2*3.14 = 6.28

KarishmaB · Answer

So we know that OC + OE = 2.4
OE is the radius of the circle (say r) and OC is made up of radius and a bit extra length. So we need to find the length of OC to get the radius of the circle.

Focus on triangle OCD (which is 90 degrees at D). 
OD is radius of the circle. DC is also radius of the circle since its length will be same as length of OF (radius of the circle).
So the triangle is isosceles right triangle. So ratio of its sides will be  1:1:\sqrt{2} 
So if the length of OD and DC is r each, the hypotenuse OC will be  \sqrt{2}r

OC + OE = 2.4
 \sqrt{2}r + r = 2.4 
 r = 1 
Circumference  = 2\pi*r = 6.3  approximately

Answer (C)

ankushbagwale · Answer

OE is C but shown here as D. Can anyone correct it, please.

Bunuel · Answer

____
Done. Thank you.

shriramvelamuri · Answer

Another explanation to add to Karishmas's (Probably Karishmas's is easier)

The points ODCF form a square with OC the diagnal. The diagnal of a square is sqrt 2 times side.

Lets the side be x and diagnal will be Sqrt 2=> X+ Sqrt 2 * X = 2.4 and you can calculate X (the radius of the circle)

Divyadisha · Answer

Draw a line from O to F. It will a square DOCF

Diagonal ^2/2= Side ^2
OC^2/2= OE ^2

OC= 2.4-OE

(2.4- OE)^2/2= OE^2

2.4-OE= \sqrt{2} *OE

2.4= \sqrt{2} OE +OE
OE= 2.4 / \sqrt{2}= 1

Circumference= 2 pi r= 2*3.14*1= 6.28

C is the answer

Manonamission · Answer

Are you using the below property of square ?

Area of a square = (Diagonal)^ 2/2

anox · Answer

Given in question

OC + OE = 2.4  --- (1)

OD = OE  --- (2)

From the figure we can also see that DCFO is a square with  side = OD

=> OC = \sqrt{2} * OD

=> OC = \sqrt{2} * OE  --- (3)

Substitute the value of OC in (1) from (3),

=> 1.41*OE + OE = 2.4 => 2.41*OE = 2.4 => OE = 1 (approx.)

We now have the radius of the circle.  OE = 1

=> C = 2 * pi * r => C = 2 * 3.14 * 1 => C = 6.3 (approx.)

(C) is the answer

Archit3110 · Answer

let part OE=OD=x
which is radius of circle 
drop a line from center O towards F ; we get a square of side x DOFC
we know that'OC+OE= 2.4 or say
OB+OC=2.4
OC is digonal of square DOFC x√2 = 2.4 -x
solve for x
x * (√2+1) = 2.4
x= 1
radius of circle is 1
circumference is 2*pi*1 ; 2*3.14*1 ; 6.28 ~6.3
option C

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.

A circle is inscribed in a right triangle as shown. If OC + OE = 2.4

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

GMAT Problem Solving (PS) Questions

A circle is inscribed in a right triangle as shown. If OC + OE = 2.4

Prep Toolkit

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