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

Answer:

The probability is about 60%

Step-by-step 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,

$AB^2=BD^2+AD^2$

$a^2=(\frac{a}{2})^2+(3r)^2$

$a^2-\frac{a^2}{4}=9r^2$

$\frac{3a^2}{4}=9r^2$

$a^2=12r^2$

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

$\text{Area of triangle=}\frac{1}{2}\times a\times 3r=\frac{1}{2}\times \sqrt{12}r\times 3r$

$=3\sqrt3 r^2$

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

In percentage:

$0.605\times 100=60.5$

which is about 60%

Hence, the correct option is A.