What is the length of side AB in equilateral triangle ABC if the

Question

https://gmatclub.com/forum/download/file.php?id=70479 What is the length of side AB in equilateral triangle ABC if the length of altitude AD to side BC is 12? A. 8√3 B. 10√2 C. 18 D. 12√3 E. 24 This is a PS Butler Question Check the links to other Butler Projects: Data Sufficiency Butler Critical Reasoning Butler Sentence Correction Butler Reading Comprehension Butler Integrated Reasoning Butler download.png

gmatophobia · Answer

\frac{AD}{BC} = \frac{AD }{ AB} = 12

In  	riangle ABD

\angle ABD = 60  
  \angle ADB = 90  
  \angle BAD = 30

AB = \frac{2}{\sqrt{3}} * AD

AB = \frac{2}{\sqrt{3}} * 12

AB = \frac{2 \sqrt{3}}{\sqrt{3}*\sqrt{3}} * AD

AB = \frac{2\sqrt{3}}{3} * 12

AB = 8\sqrt{3}

Option A

nicbo · Answer

Hey  gmatophobia  thank you for your explanation! I have two questions:

How do you get the 2/√3 in the first step?

And why do multiply by √3 in the third step?

Thank you in advance!

gmatophobia · Answer

nicbo  - Here are the responses to your queries -

In  	riangle ABD

\angle ABD = 60  
  \angle ADB = 90  
  \angle BAD = 30

Hence, the sides are on a fixed ration

BD : AD : AB = 1 : \sqrt{3} : 2

The question stem provides us with the value of AD. Hence to find the value of AB, we have to multiply by  \frac{2}{\sqrt{3}}

We had to multiply both the numerator and denominator by  \sqrt{3}  to rationalize the denominator. Note, in none of the answers presented, we have  \sqrt{3}  , hence to remove the same, we had to multiply both the numerator and denominator by  \sqrt{3}  .

Maria240895chile · Answer

let's call each side X
 \frac{AD}{BC}=12 
 \frac{AD}{x}=12

AD=12x

also the formula for height of an equilateral triangle is ((side)^2 sqrt3)/2

12x=\frac{x^2\sqrt{3}}{2}
24x=x^2\sqrt{3}

24=x\sqrt{3}  rationalize by  \sqrt{3}

\frac{24}{\sqrt{3}}=x

IMO A

\frac{24\sqrt{3}}{3} 
 =8\sqrt{3}

What is the length of side AB in equilateral triangle ABC if the

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What is the length of side AB in equilateral triangle ABC if the

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