Last visit was: 31 Oct 2024, 16:49 It is currently 31 Oct 2024, 16:49
Close
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
Your Progress

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.
Close
Request Expert Reply
Confirm Cancel
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 31 Oct 2024
Posts: 96,533
Own Kudos:
673,099
 [5]
Given Kudos: 87,883
Products:
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 96,533
Kudos: 673,099
 [5]
2
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
User avatar
nick1816
User avatar
Retired Moderator
Joined: 19 Oct 2018
Last visit: 31 Oct 2024
Posts: 1,863
Own Kudos:
6,978
 [3]
Given Kudos: 707
Location: India
Posts: 1,863
Kudos: 6,978
 [3]
2
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
User avatar
yashikaaggarwal
User avatar
Senior Moderator - Masters Forum
Joined: 19 Jan 2020
Last visit: 19 Oct 2024
Posts: 3,118
Own Kudos:
2,867
 [3]
Given Kudos: 1,510
Location: India
GPA: 4
WE:Analyst (Internet and New Media)
Posts: 3,118
Kudos: 2,867
 [3]
3
Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
NitishJain
User avatar
IESE School Moderator
Joined: 11 Feb 2019
Last visit: 24 Oct 2023
Posts: 270
Own Kudos:
Given Kudos: 53
Posts: 270
Kudos: 181
Kudos
Add Kudos
Bookmarks
Bookmark this Post
IMO D

Triangle ACB is 30-60-90 triangle
30-60-90
1: √3 : 2

Side opposite to 90* is AB= 4

Hence: BC =2
& AC = 2√3

\((Length of AC)^2\) = \((x-0)^2 \)+\( (y-0)^2\)
==> \(x^2\) + \(y^2\) =12 -----------eq 1

Also: \((Length of BC)^2\) = \((x-4)^2 \)+\( (y-0)^2\)
\((x-4)^2\) + \(y^2\) =4

Substituting \(y^2\) =12 - \(x^2\) from eq1

\(x^2\) + 16 -8x + (12 - \(x^2\)) = 4

24 = 8x
or x =3

From eq1 \(3^2\) + \(y^2\) =12
\(y^2\) = 3
y = √3 or -√3

But we know y is in +ve x quadrant and hence -√3 can be rejected

y = √3
User avatar
MayankSingh
Joined: 08 Jan 2018
Last visit: 05 Mar 2024
Posts: 289
Own Kudos:
262
 [2]
Given Kudos: 249
Location: India
Concentration: Operations, General Management
GMAT 1: 640 Q48 V27
GMAT 2: 730 Q51 V38
GPA: 3.9
WE:Project Management (Manufacturing)
Products:
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
IMO D

in triangle ABC
Sin 30 = BC/AB
=> BC = AB Sin 30 = 4 X 1/2 = 2

Cos 30 = AC/AB
=> AC = √3/2 x 4 = 2√3

Let Coordinate of C (a,b)

AC^2 = a^2 + b^2 = (2√3)^2 = 12 ...I
BC^2 = (a-4)^2 + b^2 = 4...............II

From I-II
a^2 - (a-4)^2 = 8
=> a= 3
So from I , b= √ (12-9) = √3
User avatar
GMATinsight
User avatar
GMAT Club Legend
Joined: 08 Jul 2010
Last visit: 27 Oct 2024
Posts: 6,055
Own Kudos:
14,392
 [2]
Given Kudos: 125
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Products:
Expert reply
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
Posts: 6,055
Kudos: 14,392
 [2]
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
ABC is a right triangle with angles 30º-60º-90º
i.e. ratio of Sides must be x:√3x:2x

AB = 2x = 4
i.e. AC = √3x = 2√3
and BC = x = 2

Join Point C with Center of circle O
∆OCB is an equilateral Triangle with side = BC = 2


i.e. Y-coordinate of Point C = Height of Equilateral OBC \(= (\frac{√3}{2})*2 = √3\)


and, X-coordinate of Point C = Midpoint of O and C \(= \frac{(2+4)}{2} = 3\)

i.e. Point C = (3, √3)

Answer: Option D
avatar
Superman249
Joined: 10 Jan 2018
Last visit: 04 Jan 2021
Posts: 73
Own Kudos:
133
 [1]
Given Kudos: 20
Posts: 73
Kudos: 133
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Ans is D

Two ways of doing it:-

AC = 2*sqrt(3) and BC = 2

By using Distance Formula x^2 +y^2 = Distance between point^2

Let's assume coordinate of C be x and y

We will have two-equation
x^2 + y ^2 =12
and (x-4)^2 + y^2 = 4

Solving we get y= sqrt(3)

2nd Way :- Better and easier
equation of line AC > y=x/sqrt(3) , as slope is 1/sqrt(3)
equation of BC > y=-sqrt(3)x + 4 sqrt (3)

Solving both equations to get the meeting point
we get y= sqrt(3)



Eq
User avatar
firas92
User avatar
Current Student
Joined: 16 Jan 2019
Last visit: 24 Jan 2024
Posts: 620
Own Kudos:
1,532
 [1]
Given Kudos: 142
Location: India
Concentration: General Management
GMAT 1: 740 Q50 V40
WE:Sales (Other)
Products:
GMAT 1: 740 Q50 V40
Posts: 620
Kudos: 1,532
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Note that the y coordinate of C is simply the length of perpendicular from C to AB (which is the height of the triangle with base AB)

Since \(\angleCAB\) is \(30\), the side lengths of ACB are in the ratio \(BC:AC:AB=1:\sqrt3:2\)

So since length of \(AB\) is \(4\), \(BC=2\), \(AC=2\sqrt3\)

Area of triangle ABC with base BC and height AC = Area of triangle ABC with base AB and height h (h=y coordinate of C)

\(\frac{1}{2}*AC*BC=\frac{1}{2}*AB*h\)
\(\frac{1}{2}*2\sqrt3*2=\frac{1}{2}*4*h\)

\(h=\sqrt3\)

Answer is (D)

Posted from my mobile device
User avatar
exc4libur
Joined: 24 Nov 2016
Last visit: 22 Mar 2022
Posts: 1,710
Own Kudos:
1,383
 [1]
Given Kudos: 607
Location: United States
Posts: 1,710
Kudos: 1,383
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
The figure above shows a circle whose diameter AB lies on the x-axis as shown. Triangle ACB is a right-angled triangle whose side AC makes an angle of 30 with side AB. If the coordinates of points A and B are (0,0) and (4,0) respectively, what is the y-coordinate of point C?

Sides 30-60-90 triangle
x, xV3, 2x = x, xV3, 4 = 2, 2V3, 4

Distance between points
d^2=(x1-x2)^2+(y1-y2)^2
(2V3)^2=(a-0)^2+(b-0)^2
a^2+b^2=12

(2)^2=(a-4)^2+(b-0)^2
a^2+b^2+16-8a=4
[12]+16-8a=4
a=3

(3)^2+b^2=12
b^2=3
b=V3

Ans (D)

Posted from my mobile device
avatar
vipulshahi
Joined: 24 Sep 2013
Last visit: 30 Aug 2021
Posts: 164
Own Kudos:
Given Kudos: 40
Location: Saudi Arabia
GPA: 3.8
WE:Project Management (Energy)
Posts: 164
Kudos: 110
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Given, Angle BAC = 30; AB = 4 ; Angle ACB = 90 ; Co-ordinate A=(0,0) B= (4,0)
Required = Co-ordinate C= (x,y) ; y= ??
Sol: AB = 4 ; In Tri. ABC, 2*a = 4 ; a = 2
AC = Sqt(3) * a = 2*Sqt(3) ; BC = 2
Lets calculate height , h = perpendicular drawn from point "C" to Line "AB"
h=sqt(3)
Thus, y co-ordinate of point "C" is Sqrt(3)

IMO(D)
User avatar
unraveled
Joined: 07 Mar 2019
Last visit: 31 Oct 2024
Posts: 2,712
Own Kudos:
1,978
 [1]
Given Kudos: 764
Location: India
WE:Sales (Energy)
Posts: 2,712
Kudos: 1,978
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Image
The figure above shows a circle whose diameter AB lies on the x-axis as shown. Triangle ACB is a right-angled triangle whose side AC makes an angle of 30? with side AB. If the coordinates of points A and B are (0,0) and (4,0) respectively, what is the y-coordinate of point C?


A. 1/√3

B. √3/2

C. 2/√3

D. √3

E. 2√3

The maximum y-coordinate of point C is 2 since radius of the circle is 2 i.e. the highest point from centre of the circle to the arc ACB.
So, E is out.

Triangle ACB is right angled at C. Thus, Angle ABC is 60.
From 30-60-90 triangle, sides of the triangle are in ratio as BC:AC:AB
1:√3:2 OR 2:2√3:4 in this case

Area of triangle ACB = 1/2* AC * BC = 1/2 * AB * CD (where CD is perpendicular from C to AB)
\(\frac{1}{2} * 2√3 * 2 = \frac{1}{2} * 4 * CD\)
CD = √3

Answer D.
User avatar
HoneyLemon
User avatar
Stern School Moderator
Joined: 26 May 2020
Last visit: 02 Oct 2023
Posts: 631
Own Kudos:
Given Kudos: 219
Status:Spirited
Concentration: General Management, Technology
WE:Analyst (Computer Software)
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Quote:
https://gmatclub.com/forum/download/file.php?id=55087
The figure above shows a circle whose diameter AB lies on the x-axis as shown. Triangle ACB is a right-angled triangle whose side AC makes an angle of 30? with side AB. If the coordinates of points A and B are (0,0) and (4,0) respectively, what is the y-coordinate of point C?


A. 1/√3

B. √3/2

C. 2/√3

D. √3

E. 2√3

D , imo
AB = 4 .
Since ABC is an 30-60-90 triangle
BC = 2 and AC = 2 Root (3) .

Now in triangle ABC if we draw a perpendicular CD on AB .
0.5 * AB * CD= 0.5 * BC * AC
4 * CD = 4 root (3)
CD = root (3)..

So Y coordinate of C is √3 . So D is my ans .
User avatar
monikakumar
Joined: 23 Jan 2020
Last visit: 31 Dec 2021
Posts: 236
Own Kudos:
144
 [1]
Given Kudos: 467
Products:
Posts: 236
Kudos: 144
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
30 60 90 triangle
so, 1 \(\sqrt{3}\) 2
so, 2 \(2\sqrt{3}\) 4
dist between A and C=2 =\(\sqrt{3}\) = \(\sqrt{x^2+y^2}\)
dist between B and C =2 =\(\sqrt{(x-4)^2+y^2}\)
using distance formula, and solving
we get (3,\(\sqrt{3}\))

Ans D
User avatar
Kritisood
Joined: 21 Feb 2017
Last visit: 19 Jul 2023
Posts: 504
Own Kudos:
Given Kudos: 1,091
Location: India
GMAT 1: 700 Q47 V39
Products:
GMAT 1: 700 Q47 V39
Posts: 504
Kudos: 1,127
Kudos
Add Kudos
Bookmarks
Bookmark this Post
drop a perpendicular from C to point D on AB. we need to find CD to get the coordinate for y.
CD is the height of the triangle.
This is 30 60 90 triangle AB=4 hence CB = 2 and CA = \(2 \sqrt{3}\)

\(\frac{1}{2}\)*AC*AD=\(\frac{1}{2}\)*CA*CB

4*AD=\(2*2*\sqrt{3}\)

AD=\(\sqrt{3}\)
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 35,331
Own Kudos:
Posts: 35,331
Kudos: 902
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
Moderator:
Math Expert
96533 posts