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Can anyone help with this?

 

Let PQR be an equilateral triangle, centered at O.  A point X is chosen at random inside the triangle.  Find the probability that X is closer to O than to any of the sides.  (In other words, find the probability that XO is shorter than XA, XB, and XC.)

 Mar 5, 2023
 #1
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Let's first draw a diagram to help visualize the problem:

```
          P
         / \
        /   \
       /     \
      /       \
     /         \
    Q-----------R
```

Since PQR is an equilateral triangle, we know that all the sides are equal in length and all the angles are equal to 60 degrees. Let's call the side length of PQR "s".

The condition for X to be closer to O than to any of the sides is equivalent to saying that X must lie inside the circle centered at O with radius s/3, which is the largest circle that is completely contained within PQR and centered at O. This is because the distance from O to any of the sides is s/2 (the radius of the circumcircle of PQR), and we want X to be closer to O than to any of the sides, which means the distance from X to O must be less than s/2.

Let's call this circle C. The area of C is pi times the square of its radius, which is pi times (s/3)^2 = pi*s^2/9. The area of PQR is (sqrt(3)/4)*s^2 (you can derive this formula for the area of an equilateral triangle using trigonometry). Therefore, the probability of X being inside circle C is:

Area of C / Area of PQR = (pi*s^2/9) / ((sqrt(3)/4)*s^2) = (4*pi) / (9*sqrt(3))

So the probability that X is closer to O than to any of the sides is (4*pi) / (9*sqrt(3)).

 Mar 5, 2023

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