A point is randomly put inside an equilateral triangle whose side leng

Question

A point is randomly put inside an equilateral triangle whose side length is 3 cm. What is the probability that its distance to any corner is greater than 1 ?

A.  \frac{9\sqrt{3}-1}{9\sqrt{3}}

B.  \frac{9\sqrt{3}-\pi}{9\sqrt{3}}

C.  \frac{9\sqrt{3}-2\pi}{9\sqrt{3}}

D.  \frac{9\sqrt{3}-3\pi}{9\sqrt{3}}

E.  \frac{9\sqrt{3}-2\pi}{8\sqrt{3}}

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Apo_orv · Answer

For me its C,
Hint: Area of Equi triangle - Area From 3 Corner of 60 Degree Angle(which total comes out as 180 Degree)/ Equi Tri area

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Kuanl · Answer

We first need to find the area of the triangle which is (root3)/4 x 3 x3 = (root3)/4 x 9
then as the question says pt has to be at a distance of less than or equal to 1 from one of the corners so we take a sector of 1 cm from each corner and subtract if from the area of triangle
The area of a single sector = pix1x1 x 60/360 =1/6 pi
for three corners it will be 3 x 1/6x pi= 1/2 pi
now we subtract (root3)/4x9 - 1/2 pi = (root3)9-2pi/4
the probability will be {(root3) 9 -2pi/4} / (root3)/4 x9

TestPrepUnlimited · Answer

To find this probability we need the area of the triangle without the 3 sectors divided by the area of the triangle.

We may find the area of 3 sectors divided by the area of the triangle, then subtract that from 1 for convenience.

The area of 3 sectors adds up to a semi-circle with a radius of 1, so that area of that is  \frac{\pi}{2} .

The area of the equilateral triangle is  \frac{\sqrt{3}}{4}*3^2 = \frac{9\sqrt{3}}{4} .

Then the ratio would be  \frac{2\pi}{9\sqrt{3}}  and subtract that from 1 to get C.

Ans: C

kayarat600 · Answer

Weak in geometry can anybody tell what happened with the triangle here?

Bunuel · Answer

Good news! Questions involving specific geometry knowledge are no longer tested on GMAT Focus.

Check these two topics:
 
  GMAT Syllabus for Focus Edition  
  Geometry Tested on GMAT Focus?

While specific geometry knowledge is not tested on GMAT Focus, not everything involving shapes, volumes, or areas requires specialized geometry knowledge. The area of a square or rectangle, the volume of a cube or rectangular solid, and the Pythagorean theorem are not considered specific geometry knowledge by the GMAT and can still be tested. Moreover, a question can involve shapes but test another area, such as combinations or other topics. There are several  questions involving these concepts in the GMAT Prep Focus mocks .

Some aspects of coordinate geometry are also tested on GMAT Focus under the Functions tag.

The question at hand is a hard-core geometry question, so such questions are excluded.

So, you can ignore this question.

Hope it helps.

Samin2970 · Answer

I got the part where we calculated the area of the triangle as the "total outcome" or the denominator of the probability but what did we do after? 
a) how do I determine the area of the surface within the triangle where the point can be placed? 
b) how do I determine the shape of that surface?

Bunuel · Answer

Questions involving specific geometry knowledge are no longer tested on GMAT Focus.

Check these two topics:
 
  GMAT Syllabus for Focus Edition  
  Geometry Tested on GMAT Focus?

While specific geometry knowledge is not tested on GMAT Focus, not everything involving shapes, volumes, or areas requires specialized geometry knowledge. The area of a square or rectangle, the volume of a cube or rectangular solid, and the Pythagorean theorem are not considered specific geometry knowledge by the GMAT and can still be tested. Moreover, a question can involve shapes but test another area, such as combinations or other topics. There are several  questions involving these concepts in the GMAT Prep Focus mocks .

Some aspects of coordinate geometry are also tested on GMAT Focus under the Functions tag.

The question at hand is a hard-core geometry question, so such questions are excluded.

So, you can ignore this question.

Hope it helps.

A point is randomly put inside an equilateral triangle whose side leng

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A point is randomly put inside an equilateral triangle whose side leng

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards