Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 04 Dec 2013, 21:09

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# BC=BD=DC=AD. If AB= 10, what is the length of AC?

Author Message
TAGS:
Manager
Joined: 09 Feb 2013
Posts: 121
Followers: 1

Kudos [?]: 113 [3] , given: 17

BC=BD=DC=AD. If AB= 10, what is the length of AC? [#permalink]  11 Mar 2013, 23:42
3
KUDOS
00:00

Difficulty:

45% (medium)

Question Stats:

51% (02:52) correct 48% (01:39) wrong based on 83 sessions
Attachment:

1.jpg [ 7.3 KiB | Viewed 1857 times ]
Note: figure not drawn to scale

BC=BD=DC=AD. If AB= 10, what is the length of AC?

A. 20
B. 10\sqrt{3}
C. 20\sqrt{3}/3
D. 10
E. 10\sqrt{3}/3
[Reveal] Spoiler: OA

_________________

Kudos will encourage many others, like me.
Good Questions also deserve few KUDOS.

Math Expert
Joined: 02 Sep 2009
Posts: 15058
Followers: 2516

Kudos [?]: 15433 [1] , given: 1550

Re: BC=BD=DC=AD. If AB= 10, what is the length of AC? [#permalink]  12 Mar 2013, 02:40
1
KUDOS
Expert's post
emmak wrote:
Attachment:
The attachment 1.jpg is no longer available
Note: figure not drawn to scale

BC=BD=DC=AD. If AB= 10, what is the length of AC?

A. 20
B. 10\sqrt{3}
C. 20\sqrt{3}/3
D. 10
E. 10\sqrt{3}/3

This question CAN be solved without trigonometry. In fact trigonometry is NOT tested on the GMAT, which means that EVERY GMAT geometry question can be solved without it.

Look at the figure below:
Attachment:

Triangle.png [ 18.53 KiB | Viewed 1811 times ]
Notice that triangle ABE is 30°-60°-90° right triangle (E=90° and A=30°), thus its sides are in the ratio 1 : \sqrt{3}: 2 (\sqrt{3} corresponds with AE and 2 corresponds with AB).

Now, since AE=2x+x=3x, then AE:AB=\sqrt{3}: 2 --> 3x:10=\sqrt{3}: 2 --> x=\frac{5\sqrt{3}}{3} --> AC=4x=\frac{20\sqrt{3}}{3}.

For more check Triangles chapter of Math Book: math-triangles-87197.html

Hope it's clear.
_________________
Manager
Joined: 08 Dec 2012
Posts: 67
Location: United Kingdom
Concentration: Strategy, Sustainability
Schools: LBS '16
GMAT 1: 710 Q0 V0
WE: Engineering (Consulting)
Followers: 1

Kudos [?]: 30 [1] , given: 31

Re: BC=BD=DC=AD. If AB= 10, what is the length of AC? [#permalink]  18 Mar 2013, 13:38
1
KUDOS
A step shorter than Bunuel's method

Triangle BCD is equilateral. Triangle ADB is isosceles (with angle ADB =120).

Therefore, ABC is a right angle triangle, with angle BAC=30; ACB=60 & ABC=90.

We know that a right angle triangle with 30-60-90 combination has sides of ratio 1x:\sqrt{3}x:2x, with \sqrt{3}x corresponding to AB; 1x corresponding to BC; and 2x corresponding to AC.

Since we know AB = 10, so AC = \frac{20}{\sqrt{3}} which can also be written as 20\sqrt{3}/3

Hope this makes sense.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 3732
Location: Pune, India
Followers: 801

Kudos [?]: 3165 [0], given: 136

Re: BC=BD=DC=AD. If AB= 10, what is the length of AC? [#permalink]  18 Mar 2013, 20:21
Expert's post
emmak wrote:
Attachment:
1.jpg
Note: figure not drawn to scale

BC=BD=DC=AD. If AB= 10, what is the length of AC?

A. 20
B. 10\sqrt{3}
C. 20\sqrt{3}/3
D. 10
E. 10\sqrt{3}/3

You can do it using just the Pythagorean theorem too.
Notice that you have an equilateral triangle BCD. If its side is a, its altitude BE (shown by Bunuel in the diagram) will be \sqrt{3}a/2.
Also the base AE will be a + a/2 (Since altitude of equilateral triangle bisects the base)

So 10^2 = (3a/2)^2 + (\sqrt{3}a/2)^2
100 = 12a^2/4
a = 10\sqrt{3}/3

Length of AC = 2a = 20\sqrt{3}/3
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. Veritas Prep Reviews Manager Joined: 26 Feb 2013 Posts: 54 Concentration: Strategy, General Management GMAT 1: 660 Q50 V30 WE: Consulting (Telecommunications) Followers: 0 Kudos [?]: 1 [0], given: 16 Re: BC=BD=DC=AD. If AB= 10, what is the length of AC? [#permalink] 19 Mar 2013, 08:13 From BC=BD=DC, we know angle DBC= 60 and Angle BDA=120. And AD=AB so angle ABD=30. Therefor triangle ABC is right angle trangle with angle B=90 and Angle A=30. now Cos(A)= AB/AC=10/AC=Sqrt3/2 hence AC=20/srt3 or (20*srt3)/3 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 3732 Location: Pune, India Followers: 801 Kudos [?]: 3165 [0], given: 136 Re: BC=BD=DC=AD. If AB= 10, what is the length of AC? [#permalink] 27 Mar 2013, 19:51 Expert's post rakeshd347 wrote: Hi There I understood your approach but I solved this problem with another approach which is right but I am getting the different result. Can you tell me what is wrong with this approach. angle BDC,BCD and DBC are all 60 degrees as it is equilateral triangle. Now in the triangle BAD angle BDA will be 120 because of straight line. so the rest of the two angles will be 30 each which makes the angle ABC a rectangle. So the triangle ABC is rectangle and if you solve it with this approach you will get the answer 10multiply by root3/3 Can you please tell me whats wrong with this. Thanks You are right that triangle ABC is right angled at B. Also, AC = 2*BC, AB = 10 BC^2 + 10^2 = AB^2 = (2BC)^2 10^2 = 3(BC)^2 BC = 10\sqrt{3}/3 But the question asks you the length of AC. AC = 2*BC = 20\sqrt{3}/3 _________________ Karishma Veritas Prep | GMAT Instructor My Blog Save$100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

Manager
Joined: 28 Apr 2013
Posts: 72
Location: India
GPA: 3.6
WE: Medicine and Health (Health Care)
Followers: 0

Kudos [?]: 9 [0], given: 50

Re: BC=BD=DC=AD. If AB= 10, what is the length of AC? [#permalink]  25 Nov 2013, 06:45
emmak wrote:
Attachment:
1.jpg
Note: figure not drawn to scale

BC=BD=DC=AD. If AB= 10, what is the length of AC?

A. 20
B. 10\sqrt{3}
C. 20\sqrt{3}/3
D. 10
E. 10\sqrt{3}/3

Another way; since BD=BC=DC, triangle BDC equilateral triangle hence angle BCA = 60 degrees. it mens that other angles angle ABC + angle BAC = 180-60 = 120. Which means that AB = < AC. Which means AC >=10. Let BC = X ; Therefore AC= 2X.
Using pythagoreus theorem AC^2 = AB^2 + BC^2 = 100 +x
4x^2 = 100 + x^2
3x^2 = 100
x = 10[square_root]3/3
AC = 2X = 20 [square_root]3/3

Re: BC=BD=DC=AD. If AB= 10, what is the length of AC?   [#permalink] 25 Nov 2013, 06:45
Similar topics Replies Last post
Similar
Topics:
1 If B is the midpoint of AC, what is the length of BE? 3 15 Nov 2010, 11:00
What is the length of the chord AB 4 29 Jan 2012, 22:52
5 In the diagram, what is the length of AB? 14 05 Feb 2012, 16:30
4 The side lengths of triangle ABC are such that AC > BC > AB. 7 11 Mar 2013, 14:13
1 What is the length of AB ? 1 10 Aug 2013, 06:57
Display posts from previous: Sort by