Question

A circle is inscribed in an equilateral triangle. A point in the figure is selected at random. Find the probability that the point will be in the shaded region.

A. about 60%

B. about 50%

C. about 75%

D. about 30%

Answer 1

It’s gotta be A because the circle is usually 60%

Answer 2

The probability is about 60%

Explanation:

Given a circle is inscribed in an equilateral triangle.

A point in the figure is selected at random then we have to find its probability.

Let the radius of circle is r and side of equilateral triangle is a

OD=r

As centroid of the triangle divides the median into 2:1

∴ AO=2r

In ΔABD,

`AB^2=BD^2+AD^2`

`a^2=((a)/(2))^2+(3r)^2`

`a^2-(a^2)/(4)=9r^2`

`(3a^2)/(4)=9r^2`

`a^2=12r^2`

`\text{Area of circle=}\pi r^2`

`\text{Area of triangle=}(1)/(2)* a* 3r=(1)/(2)* √(12)r* 3r`

`=3\sqrt3 r^2`

`Probability=\frac{\text{area of circle}}{\text{area of triangle}}=(\pi r^2)/(3\sqrt3 r^2)=0.604599788078\sim 0.605`

In percentage:

`0.605* 100=60.5`

which is about 60%

Hence, the correct option is A.