It is currently 25 Oct 2020, 19:40

### 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

# Triangles ABC, ACD, and ABD are all isosceles triangles. Poi

Author Message
TAGS:
Founder
Joined: 18 Apr 2015
Posts: 13673
Followers: 305

Kudos [?]: 3531 [0], given: 12627

Triangles ABC, ACD, and ABD are all isosceles triangles. Poi [#permalink]  07 May 2020, 10:34
Expert's post
00:00

Question Stats:

70% (01:47) correct 29% (01:59) wrong based on 17 sessions
Attachment:

#greprepclub Triangles ABC, ACD, and ABD are all isosceles triangles..jpg [ 17.03 KiB | Viewed 909 times ]

Triangles ABC, ACD, and ABD are all isosceles triangles. Point E (not shown) is the midpoint of $$\overline{BD}$$. If the ratio of the length of $$\overline{CE}$$ to the length of $$\overline{BC}$$ is equal to $$\sqrt{3} : 2$$, then what is the measure, in degrees, of ∠CAD?

A. 10

B. 15

C. 30

D. 45

E. 60

Kudos for the right answer and explanation
[Reveal] Spoiler: OA

_________________

- Do you want all the GRE Prep Club resources at a glance (sort of ) ?? Look at GRE Prep Club Official LinkTree page
- Do you want all the GRE Prep Club FREE resources in one spot ?? Look at Free GRE Materials - Where to get it!! (2020)
- Do you want all the FREE GRE tests on earth ?? Look at FREE GRE Practice Tests [Collection] - New Edition (2021)
- Do you want to know the GRE score in the Top 25 BS in 7 continents ?? Look at here The GRE average score at Top 25 Business Schools 2020 Ed.
- Need Practice 20 Free GRE Quant Tests available for free with 20 Kudos
- GRE Prep Club Members of the Month: Each member of the month will get three months free access of GRE Prep Club tests.

Moderator
Joined: 16 Oct 2019
Posts: 63
Followers: 0

Kudos [?]: 80 [1] , given: 22

Re: Triangles ABC, ACD, and ABD are all isosceles triangles. Poi [#permalink]  07 May 2020, 20:11
1
KUDOS
△ABC, △ACD and △ABD are isosceles;
In △ABC: AC=BC, ∠BAC = ∠ABC = x
△ACD: AC=CD, ∠CAD = ∠CDA = x

Now, In △BCE, CE/BC = √3/2
sin ∠CBE = length of oppo site/ length of hypotenus
CE / BC = √3/2 = Sin 60°
hence, ∠CBE = 60°
as BC = CD; ∠CDE = 60°

In △ABD; ∠A + ∠B + ∠C = 180
2x + (x+60) + (x+60) = 180
4x = 60
x = 15
Attachments

#greprepclub Triangles ABC, ACD, and ABD are all isosceles triangles..jpg [ 12.99 KiB | Viewed 846 times ]

Intern
Joined: 30 Aug 2020
Posts: 19
Followers: 0

Kudos [?]: 16 [1] , given: 3

Re: Triangles ABC, ACD, and ABD are all isosceles triangles. Poi [#permalink]  12 Oct 2020, 07:47
1
KUDOS
The correct answer is b) 15 degrees.

Now, let us work our way from the ratio CE/BC = √3:2 = > ECB is a 30-60-90 triangle.
=> Angle CBD = 60;
Angle CDB = 60
Now, notice that since two angles are 60 degree in Triangle CDB, CDB is equilateral
=> Angle DCB = 60 degree

Now notice that CE bisects the angle DCB

Hence, ACB + BCE = 180
=> ACB + 30 = 180
=> ACB = 150

Similarly,
ACD + DCE = 180
=> ACD + 30 = 180
=> ACD = 150

Now notice, that equal sides of the isosceles triangle DAC must be the sides AC and CD that are adjacent to angle ACD
(This is because if side opposite to ACD is the equal side, then 150 + 150 > 180 degrees. Hence not possible)

This ,means that CAD = CDA = x
=> x + x = 150 = 180
=> 2x = 30
=> x = 15
Re: Triangles ABC, ACD, and ABD are all isosceles triangles. Poi   [#permalink] 12 Oct 2020, 07:47
Display posts from previous: Sort by