+54 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.
+551 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.
+868 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.
