A 30-60-90 triangle is inscribed in a circle. The length of the hypotenuse is 12 inches. If a coin is tossed on the figure, what is the probability that the coin will land in the circle, but outside the triangle?
All right-triangles inscribe in a circle with a diameter equal to the hypotenuse.
Therefore for the 30-60-90 triangle, the radius of the circle is 6 inches, and the short side is also 6 inches. The height is 6sqrt(3), so the area of the triangle is
At=36sqrt(3)/2 = 18 sqrt(3) sq.in.
The area of the circle is
Ac=π6^2=36π
The probability of falling inside the circle and outside the triangle is therefore
P(C\T)=(Ac-At)/Ac
the quesiton is land in the circle, but outside the triangle. Is it the probability is (AC - AT)/AC
81.22/113 x 100%
= 72%
Correct.
I get
P(C\T) (probability inside circle minus triangle)
=(Ac-At)/Ac
=81.92/113.1=72.4%
To find the probability that the coin will land in the circle but outside the triangle, we need to compare the area of the circle to the area of the triangle.
Let's start by finding the area of the circle. The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius. Since the radius of the circle is half of the hypotenuse, which is 12 inches, the radius is 6 inches.
So, the area of the circle is A = π(6)^2 = 36π square inches.
Now, let's find the area of the triangle. The area of a triangle can be calculated using the formula A = 0.5(base)(height). In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is √3 times the length of the shorter leg.
In this case, the hypotenuse has a length of 12 inches, which means the shorter leg is 12/2 = 6 inches, and the longer leg is √3 * 6 = 6√3 inches.
The base of the triangle is the shorter leg, which is 6 inches. The height of the triangle can be found using the Pythagorean theorem, where a^2 + b^2 = c^2. In this case, a = 6 inches (height), b = 6√3 inches (longer leg), and c = 12 inches (hypotenuse).
Using the Pythagorean theorem, we can find the height:
(6)^2 + (6√3)^2 = (12)^2
36 + 108 = 144
144 = 144
The height is found to be 12 inches.
Using the area formula, the area of the triangle is A = 0.5(6)(12) = 36 square inches.
To find the probability, we divide the area of the circle outside the triangle by the total area of the circle:
Probability = (Area of Circle outside Triangle) / (Total Area of Circle)
Probability = (36π - 36) / (36π)
Simplifying the expression, we have:
Probability = (π - 1) / π
So, the probability that the coin will land in the circle but outside the triangle is (π - 1) / π, where π is approximately 3.14159.