BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

Geometry Question (Arc, line segement and right triangles)

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Senior | Next Rank: 100 Posts
Posts: 36
Joined: Mon Jan 28, 2008 4:12 pm
Location: Montreal, Canada
Thanked: 2 times

Geometry Question (Arc, line segement and right triangles)

by Kaunteya » Fri Feb 29, 2008 8:18 am
The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is (4 x Pi)/3, what is the length of line segment RU?

A. 4/3
B. 8/3
C. 3
D. 4
E. 6

Okay, so in this question I have figured out that the arc RTU is 1/6 of the Circumference, or 60 degrees. That means that the opposite angle in the triangle from the arc is 30 degrees, making it a 90-30-60 triangle. So if the length of the hypoteneus is 8 (4x2), how can I get the length of line segment RU using Pythagoras. You still have two unknowns. They claim that the answer is E. I can only logically choose D. There is no explination how they got to the number 6. I can't seem to get it because the line across the 60 degree angle should be greater than the line segment across the 30 degree angle (and we are looking for the line segment distance across the 30 degree angle). Please explain?

Kaunteya
Top
Source: — Problem Solving |
Master | Next Rank: 500 Posts
Posts: 111
Joined: Thu Jan 31, 2008 4:05 pm
Thanked: 18 times
Followed by:1 members

by xilef » Fri Feb 29, 2008 3:33 pm
hmm, I'm also having problems with this, well since the arc RTU is 1/6 of the circumference, then the opposite angle of segment RU is 60, which makes it an equilateral triangle ROU (O being the origin) so RU = 4 ???
Top
User avatar
GMAT Instructor
Posts: 3225
Joined: Tue Jan 08, 2008 2:40 pm
Location: Toronto
Thanked: 1710 times
Followed by:614 members
GMAT Score:800

by Stuart@KaplanGMAT » Fri Feb 29, 2008 6:30 pm
The mistake you made is with the triangle you created.

If the arc is 1/6 the circumference, then the interior angle will be 1/6 of 360, or 60 degrees.

Both exterior points of the triangle are on the circumference of the circle, so lines OR and OT are both radii of the circle. Every triangle formed by two radii will be isoscles, and in this case since we have an interior angle of 60 we actually have an equliateral (i.e. 60/60/60) triangle.

So, line RT will be equal to the radius of the circle, which would make the answer 4.

If 4 isn't the answer, then either the question is misquoted or it's not 1/6th of the circle.
Image

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

Kaplan Exclusive: The Official Test Day Experience | Ready to Take a Free Practice Test? | Kaplan/Beat the GMAT Member Discount
BTG100 for $100 off a full course
Top
Senior | Next Rank: 100 Posts
Posts: 83
Joined: Mon Oct 22, 2007 4:44 am
Thanked: 1 times

by its_me07 » Sat Mar 01, 2008 12:28 am
How arc RTU = 1/6 of the circumfrence

plz explain.
Top
Senior | Next Rank: 100 Posts
Posts: 74
Joined: Tue Sep 25, 2007 7:58 am
Thanked: 4 times

by joshi.komal » Sat Mar 01, 2008 12:39 am
its_me07 wrote: How arc RTU = 1/6 of the circumfrence
Arc /circumference = (4* pi/3) / (2pi * 4) [Given radius = 4 ]
its_me07 wrote: I think...as RU is a line segment touching two ends of the circle ...it would be greater than the radius and less the diameter .
There is no rule as to why the this line segment should be greater than radii in fact it will depend on the angle the triangle forms at the center since the triangle formed in this manner will be an isosceles triangle (2 sides are the radii)

Hope it is clear now.

Thanks
Komal
Top
Legendary Member
Posts: 941
Joined: Sun Dec 27, 2009 12:28 am
Thanked: 20 times
Followed by:1 members

by bhumika.k.shah » Fri Jan 08, 2010 9:09 pm
Kaunteya wrote:The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is (4 x Pi)/3, what is the length of line segment RU?

A. 4/3
B. 8/3
C. 3
D. 4
E. 6

Okay, so in this question I have figured out that the arc RTU is 1/6 of the Circumference, or 60 degrees. That means that the opposite angle in the triangle from the arc is 30 degrees, making it a 90-30-60 triangle. So if the length of the hypoteneus is 8 (4x2), how can I get the length of line segment RU using Pythagoras. You still have two unknowns. They claim that the answer is E. I can only logically choose D. There is no explination how they got to the number 6. I can't seem to get it because the line across the 60 degree angle should be greater than the line segment across the 30 degree angle (and we are looking for the line segment distance across the 30 degree angle). Please explain?

Kaunteya
How did u get the arc as 1/6th of the Circumference ? Can u explain step by step ...
Top
User avatar
Master | Next Rank: 500 Posts
Posts: 124
Joined: Tue Dec 22, 2009 8:32 am
Location: Karjat
Thanked: 3 times

by linkinpark » Sat Jan 09, 2010 8:39 am
circle's radius = 4, so circumference = 2*pi*r = 8pi = length of full arc for sake of simplicity, and in terms of degrees its 360 degrees

now we are told length of arc RTU = 4/3pi but based on formula for arc's length it is as follows

[measure of angle /360] * 2*pi *r from this
length of arc RTU = 4/3*pi = angle /360 * 8*pi, solving it you'll get angle formed at two radii OR and OT [assuming center = o] = 60 degrees which is also 1/6 th part of whole circle [angle/360]
attached is diagram [pardon its bad :D]

hope its clearImage[/img]
Top
Post new topic Post Reply
• Page 1 of 1