In the given isosceles triangle, if AB = BC = 2 and x = 30°, what is t

Question

In the given isosceles triangle, if AB = BC = 2 and x = 30°, what is the area of the triangle?

A. √3/2
B. 1
C. √3
D. 2
E. 2√3

BrentGMATPrepNow · Accepted Answer

Aside: Although we can, indeed, solve this question using trigonometric ratios, the GMAT test makers don't expect students to know about trigonometric ratios. So another approach is to apply what we know about special 30-60-90 right triangles

Since we're told that AB = BC, we know that the triangle is an  isosceles  triangle, which means ∠BCA =   30°   
So if we draw in the altitude from point B, we get two identical right triangles. 
 https://i.imgur.com/8pMSYVE.png

Also notice that these right triangles are both  special 30-60-90 right triangles 
So let's compare these triangles with our   base 30-60-90 right triangle  
 https://i.imgur.com/etxezBf.png

When we do this, we see that the two   red right triangles   are  identical  to the   base triangle  
So, we can add the following measurements to our diagram:
 https://i.imgur.com/g8ndpOG.png

At this point we're ready to find the area of the triangle.

Area of triangle = (base)(height)/2

So, the area  = \frac{(\sqrt{3}+\sqrt{3})(1)}{2}= \frac{(2\sqrt{3})(1)}{2}=\sqrt{3}

Answer: C

Cheers,
Brent

ArunSharma12 · Answer

Img.png let the side AC be x. \frac{2}{sin(30)}=\frac{x}{sin(120)} x = \frac{2*sin(120)}{sin(30)} x = \frac{2*cos(30)}{sin(30)} = \frac{2*2*\sqrt{3}}{2} ; sin(90+x)=cos(x) x = 2*\sqrt{3} To find area we can use formula A =\frac{1*a*b*sin(x)}{2} ; where a,b are two adjacent sides with angle x between them A =\frac{1*2*2*\sqrt{3}*sin(30)}{2} A =\sqrt{3} Ans: C

GMATinsight · Answer

Drop a perpendicular BD from point B on AC

Now we have two 30-60-90 right triangles with hypotenuse 2 unit long

the length of BD = 1
AD = √3

i.e. AC = 2√3

Area = (1/2)*AC*BD = (1/2)*2√3*1 = = √3

Answer: Option C

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In the given isosceles triangle, if AB = BC = 2 and x = 30°, what is t

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In the given isosceles triangle, if AB = BC = 2 and x = 30°, what is t

Prep Toolkit

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