Triangle ABC is equilateral with side length 3. A point x is randomly chosen within ABC . What is the probability that X is no more than 1 unit away from vertex A?

Lightning
Mar 29, 2018

#1**+1 **

The area of the triangle is

(1/2) (3)^2 √3/2 = (9/4)√3 units^2

The area that is no more than unit from A is (1/6) of a circle with a center at A and a radius of 1 =

pi *(1/6)(1)^2 = pi / 6 units' 2

So....the probability that X is no more than 1 unit from A is

[pi / 6 ] / [(9/4)√3 ] =

4 pi / [ 54√3] =

2pi / [27√3 ] ≈ 0.134 ≈ 13.4 %

CPhill
Mar 29, 2018