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Triangle DEF is inscribed in circle G. In circle G,  angle EDF=29 and arc DF has a measure of 132. If the circumference of circle G is 24 units, what is the length of arc DE? Express your answer as a common fraction.

 May 27, 2024
 #1
avatar+385 
+2

We are given that the circumference of circle G is 24 units, so the radius of circle G is 24/2π=12/π units.

 

An inscribed angle intercepts an arc that has a measure half that of the central angle that subtends the same arc [Inscribed Angle Theorem]. Therefore, central angle ∠EDF measures 2⋅29∘=58∘.

 

 

The measure of arc DF is given as 132 degrees. Since the sum of the degrees around a circle is 360∘, the measure of arc DE is 360∘−58∘−132∘=170∘.

 

The ratio between the arc's central angle θ and 360∘ is equal to the ratio between the arc length s and the circle's circumference c [Arc Length Ratio Theorem].

 

In this case, the ratio of the arc DE's central angle θ (which is 170∘ ) to 360∘ is equal to the ratio of the arc DE's length s (what we want to find) to the circle's circumference c (which is 24 units).

 

Therefore:

 

360∘θ​=cs​

 

360∘170∘​=24 unitss​

 

Solving for s, we get the arc length s of arc DE:

 

s=360∘170∘​⋅24 units = 17/3 units

 

Therefore, the answer is 17/3.

 May 27, 2024
 #2
avatar+620 
+2

Since angle EDF is inscribed in a circle, we know that the measure of the central angle subtended by arc DF is double the measure of angle EDF. Therefore, the central angle for arc DF is 29 x 2 = 58 degrees.

Since the circumference of the circle is 24 units and the central angle for arc DF is 58 degrees, we can use the formula for the circumference of a circle to find the radius of circle G.

C = 2πr
24 = 2πr
r = 12/π

Now, we can use the formula for the length of an arc, given by:

Length of arc = (central angle/360) * circumference

For arc DE, the central angle is 360 - 132 - 58 = 170 degrees.

Length of arc DE = (170/360) * 2π(12/π)


Length of arc DE = (17/36) * 24


Length of arc DE = 8/3 units

Therefore, the length of arc DE is 8/3 units.

 May 27, 2024

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