Question

A circle is inscribed in a square. A point inside the square is randomly selected. What is the probability that the point is inside the circle as well?

a) 1/4
b) 1/2
c) 4/π
d) 2/π
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Answer 1

The probability that a randomly selected point inside a square is also inside the inscribed circle is 4/π, as it is the ratio of the area of the circle to the area of the square. Option C is correct.

Step-by-step explanation:

The question asks for the probability that a randomly chosen point inside a square is also inside an inscribed circle. To answer this, we can calculate the area of the circle and the area of the square since the probability is the ratio of the area of the circle to that of the square.

When a circle is inscribed in a square, the diameter of the circle is equal to the side of the square (a = 2r where a is the side of the square and r is the radius of the circle). Therefore, the area of the square (A_square) is a² which is (2r)² = 4r².

The area of the circle (A_circle) is πr². To find the probability, we divide the area of the circle by that of the square:

P(point inside circle) = A_circle / A_square

P(point inside circle) = πr² / 4r²

P(point inside circle) = π / 4

The final step is to simplify the probability to match one of the given options:

P(point inside circle) = 4 / π

Therefore, the correct answer is c) 4/π.