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# help

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A dart is thrown at a random point on a circular dartboard with a radius of 8 cm. What is the probability that the dart lands in the shaded region? Answer in terms of π.

Apr 28, 2020

#1
+631
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Angle A subtends to arc BC, which is equal to 180. Therefore angle A is 180/2 = 90. Knowing that it is a 45-45-90 triangle, can you figure out the area of the triangle? After that, just subtract it from the area of the semi-circle.

Apr 28, 2020
#2
+485
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By Thales' Theorem: angle BAC = 90 degrees(it subtends an arc of 180 degrees)

Because we know that, we know that triangle BAC is a right triangle with angle measurements of 45, 45, 90

Next, realize that the hypotenuse of said triangle is the diameter BC, which has length 16( which is 8 *2).

Now that we have our hypotenuse, realize that the ratio of a 45-45-90 triangle is

x, x, $$x\sqrt{2}$$

That means that our leg of our right triangle is equal to:

$$16\over{\sqrt{2}}$$$$16\sqrt{2}\over2$$ by rationalizing the denominator(multiplying it by $$\sqrt{2}\over\sqrt{2}$$)

This equals

$$8\sqrt{2}$$

Triangle area formula is 1/2 * b * h

Since the base and height are the same in this scenario, we have that the area of triangle BAC is:

$$(8\sqrt{2})^2/2 = 64*2/2 = 64$$

The area of the entire circle is 64pi, so half of it is 32pi. Our area of the shaded region is then:

$$32\pi-64$$

The probability is then this part divided by the entire area of the dart board, which is 64pi, so our answer is:

$$32\pi-64\over64\pi$$

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Apr 28, 2020
#3
+23646
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Area of circle = pi r^2    =    64 pi

subtract 1/2 (unshaded area) = 32 pi

subtract area of triangle  1/2 b h = 1/2 16 8 = 64

(32 pi - 64) / (64 pi)

Apr 28, 2020