Question

1. 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. about 40% about 25% about 60% about 50%

Explanation:

Step1: Recall area formulas

Let the side length of the equilateral triangle be \(a\). The area of an equilateral triangle \(A_{\triangle}=\frac{\sqrt{3}}{4}a^{2}\). For a circle inscribed in an equilateral triangle, the radius \(r = \frac{\sqrt{3}}{6}a\). The area of the circle \(A_{\odot}=\pi r^{2}=\pi(\frac{\sqrt{3}}{6}a)^{2}=\frac{\pi}{12}a^{2}\)

Step2: Calculate the probability

The probability \(P=\frac{A_{\odot}}{A_{\triangle}}\). Substitute the area formulas: \(P = \frac{\frac{\pi}{12}a^{2}}{\frac{\sqrt{3}}{4}a^{2}}=\frac{\pi}{3\sqrt{3}}\approx\frac{3.14}{3\times1.732}\approx 0.604\approx 60\%\)

Answer:

about \(60\%\)