Core Connections Integrated II, 2015
CC
Core Connections Integrated II, 2015 View details
Chapter Closure

Exercise 118 Page 592

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.

a circle with an 8 cm long chord. This cord makes an arch of 120 degrees.
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^(∘).
a circle with an 8 cm long chord. This cord makes an arch of 120 degrees. Two radius with endpoints on each end of the chord make a triangle and make an angle of 120 degrees at the center.

Let's isolate the isosceles triangle and draw the height of it. We then create two right triangles.

An isosceles triangle where the longer side has a length of 8. Divided into two 30-60-90 right triangles with base 4.

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.

a circle with an 8 cm long chord

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.