A square is inscribed in a circle. A point in the figure is selected at random. Find the probability that the point will be in the shaded region.

Let x = side length of the square and d = diameter of the circle
d also is the hypotenuse of the square

Through the pythagorean theorem, we know that
x^2 + x^2 = d^2
2x^2 = d^2

Because d = 2r, this means
2x^2 = d^2
2x^2 = (2r)^2
2x^2 = 4r^2
x^2 = 2r^2

The area of the square is x^2, which is the same as 2r^2 as shown above

The area of the circle is A = pi*r^2

Divide the area of the square over the area of the circle to get
(x^2)/(pi*r^2) = (2r^2)/(pi*r^2) = 2/pi

The exact answer, in terms of pi, is the fraction 2/pi

If you want an approximate answer, then use a calculator to get roughly 0.6366 which is about 63.66%

priyankapandeyVT priyankapandeyVT · Answer 2 · 2022-07-25T06:58:03+02:00

The probability of the point will be in the shaded region is 0.64 or 64%.

A square is inscribed in circle. Probability of point to be pick from the square to be find out.

What are Venn diagram?

Venn diagram are the Figures of probabilities to understand better.

Here, diameter of circle = d and radius r =d/2, edge of the square = a
By Pythagorean theorem,
a²+a²=d²
2a²= (2r)²
a²=2r²
Area of the square =2r²
Area of the circle = πr²
Now probability = Area of the square/Area of the circle
= 2r²/πr²
= 2/π
= 0.64

Thus, The probability of the point will be in the shaded region is 0.64 or 64%.

Learn more about Venn diagram here:
https://brainly.com/question/1605100

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