Last visit was: 20 Jul 2024, 01:57 It is currently 20 Jul 2024, 01:57
Toolkit
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

# Let S be a point on a circle whose center is R. If PQ is a chord that

SORT BY:
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 94421
Own Kudos [?]: 642442 [12]
Given Kudos: 86332
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11476
Own Kudos [?]: 34457 [9]
Given Kudos: 322
General Discussion
GMAT Club Legend
Joined: 08 Jul 2010
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Posts: 6020
Own Kudos [?]: 13810 [0]
Given Kudos: 125
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Director
Joined: 21 Jun 2017
Posts: 632
Own Kudos [?]: 540 [0]
Given Kudos: 4092
Location: India
Concentration: Finance, Economics
GMAT 1: 660 Q49 V31
GMAT 2: 620 Q47 V30
GMAT 3: 650 Q48 V31
GPA: 3.1
WE:Corporate Finance (Non-Profit and Government)
Let S be a point on a circle whose center is R. If PQ is a chord that [#permalink]
Any other way ?
Also, why is this not a square. Diagnols bisect at 90 in a square as well

edited: answer to myself rhombus also has bisectors at 90 degrees

Posted from my mobile device

Originally posted by ShankSouljaBoi on 31 Oct 2018, 20:13.
Last edited by ShankSouljaBoi on 25 Nov 2019, 04:31, edited 1 time in total.
Intern
Joined: 18 Jul 2018
Posts: 5
Own Kudos [?]: 1 [0]
Given Kudos: 15
Re: Let S be a point on a circle whose center is R. If PQ is a chord that [#permalink]
Bunnel, could you please explain? Thanks so much!
Director
Joined: 21 Jun 2017
Posts: 632
Own Kudos [?]: 540 [1]
Given Kudos: 4092
Location: India
Concentration: Finance, Economics
GMAT 1: 660 Q49 V31
GMAT 2: 620 Q47 V30
GMAT 3: 650 Q48 V31
GPA: 3.1
WE:Corporate Finance (Non-Profit and Government)
Re: Let S be a point on a circle whose center is R. If PQ is a chord that [#permalink]
1
Kudos
Please have a reference of the above figure posted by GMATinsight for the solution i am posting right now.

We know that radius of circle is OB and OC both.
Pick triangles OCS and SCB , they are congruent by SAS (CS=CS Angle = 90 and OS = SB given)
---> OC= CB
or OC = OB = CB ---> Equilateral triangle . Hence angle COA is 60 + 60 = 120 (Tirangle OAB is also equilateral by same reeasoning as above).

Now (120/360*2pi*OC)/2piOC = required answer = 1/3.

Current Student
Joined: 10 Sep 2019
Posts: 137
Own Kudos [?]: 33 [0]
Given Kudos: 59
Location: India
Concentration: Social Entrepreneurship, Healthcare
GMAT 1: 680 Q49 V33
GMAT 2: 720 Q50 V37
GRE 1: Q167 V159
GPA: 2.59
WE:Project Management (Non-Profit and Government)
Re: Let S be a point on a circle whose center is R. If PQ is a chord that [#permalink]
Since the chord PQ is the purpendicular bisector of SR (which is the radius r), it divides it into r/2 & r/2.

This forms 2 triangles with hypotenuse r and height r/2. Using properties of triangles (& trigonometry), CosX = Adj/hyp

=> CosX = 1/2 => x= 60 degrees. => total angle is 120 degrees.

=> the arc length is 120/360 times the total circumference = 1/3

Ans B
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6804
Own Kudos [?]: 30839 [1]
Given Kudos: 799
Re: Let S be a point on a circle whose center is R. If PQ is a chord that [#permalink]
1
Kudos
Top Contributor
Bunuel wrote:
Let S be a point on a circle whose center is R. If PQ is a chord that passes perpendicularly through the midpoint of RS, then the length of arc PSQ is what fraction of the circle`s circumference?

A. 1/π

B. 1/3

C √3/(π+2)

D. 1/(2√2)

E. 2√3/(3π)

Here's what the diagram looks like:

From here let's add lines from the center (R) to P and Q.
At the same time, let's say the radius of the circle is 2, which means we get the following measurements:

We now have two small right triangles in our diagram.
Since we know the length of two of the three sides, we can apply the Pythagorean theorem to find the length of the third sides:

Notice that the two right triangles have the lengths 1, 2, and √3, which are the lengths of the base 30-60-90 right triangle.
This means we add the following angles to our diagram:

We can now see that angle PRQ = 120°
The entire circumference of the circle encompasses an angle of 360°

So the fraction of the circumference occupied by arc PSQ = 120°/360° = 1/3

Manager
Joined: 03 May 2020
Posts: 108
Own Kudos [?]: 36 [1]
Given Kudos: 512
Let S be a point on a circle whose center is R. If PQ is a chord that [#permalink]
1
Kudos
*midpoint* is the keyword as it indirectly leads us to the ratio of the sides of isocèles triangles.in the figure attached we can deduce that sides are in the ratio:radius:radius/2, or in the ratio x:2x ( this is a 30-60-90) isocèles triangle), and the angles formed at the circumference of the circle is 30(this 30 is formed on both ends of the arc so the inscribed angle is 30+30 . This is the inscribed angle, so according to central angle theorem, angle formed at the centre is twice the inscribed angle so 60*2=120, this is 120/360 of the the circumference of the entire circle.
Attachments

File comment: Pfa

E493E075-D8DB-40A1-8DA8-98BB49543F8F.jpeg [ 496.37 KiB | Viewed 1410 times ]

Let S be a point on a circle whose center is R. If PQ is a chord that [#permalink]
Moderator:
Math Expert
94421 posts