Chapter Closure
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Connect the ends of the arc with the center of the circle. This creates an isosceles triangle with a base of 8 inches and a vertex angle of 120^(∘).
≈ 9.2 inches
We are told that a knife sitting on a plate creates a minor arc with the knife's ends on the edges of the plate. Let's illustrate the described situation.
We can connect the arc's endpoints with a pair of radii of the circle. This forms an isosceles triangle with a vertex angle of 120^(∘).
Let's isolate the isosceles triangle and draw the height of it. We then create two right triangles.
As we can see, the radius of the circle is also the hypotenuse of a 30-60-90 triangle. In such a triangle, the hypotenuse is twice the length of the shorter leg, and the longer leg is sqrt(3) times greater than the hypotenuse. Since the longer leg is 4 inches, the shorter leg must be 4sqrt(3) and the hypotenuse must be 2( 4sqrt(3)). Let's add this information to the diagram.
When we know the radius, we can find the diameter by multiplying the radius by 2. 2(2( 4sqrt(3)))=16/sqrt(3)≈ 9.2 in.