Answer :
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Answer:
C. 159°
Step-by-step explanation:
The exterior angle at B is half the difference of the measures of the arcs it intercepts:
(3x +19)° = 1/2((17x -3)° -91°)
6x +38 = 17x -94 . . . . . . . . . . multiply by 2, divide by °
132 = 11x . . . . . . . . . . . . . add 94-6x
x = 12 . . . . . . . . . . . . divide by 11
Then long arc AD is ...
arc AD = (17(12) -3)° = 201°
Arc DCA is the rest of the circle:
arc DCA = 360° -201° = 159°
The measure of arc DCA in the circle is 159 degrees.
The correct option is C) 159 degrees.
What is the measure of arc DCA?
To determine the measure of arc DCA, first, we find the value of x:
Using angle-arc relationship:
External angle = 1/2 × ( Major arc - Minor arc )
Plug in the values from the figure:
( 3x + 19 ) = 1/2 × ( (17x - 3 ) - 91 )
Solve for x:
3x + 19 = 1/2 × ( 17x - 3 - 91 )
3x + 19 = 1/2 × ( 17x - 94 )
2(3x + 19) = ( 17x - 94 )
6x + 38 = 17x - 94
17x - 6x = 38 + 94
11x = 132
x = 132/11
x = 12
Now, we find the measure of arc DA;
Arc DA = ( 17x - 3 )
Plug in x = 12:
Arc DA = 17(12) - 3
Arc DA = 204 - 3
Arc DA = 201 degrees
Now, we find arc DC:
Arc DC = 360 - Arc CA - Arc DA
Arc DC = 360 - 91 - 201
Arc DC = 68
Now, the measure of arc DCA will be:
Arc DCA = arc DC + arc CA
Arc DCA = 68 degrees + 91 degrees
Arc DCA = 159°
Therefore, arc DCA measures 159 degrees.
Option C) 159° is the correct answer.
Learn about Intersecting secants theorem here: https://brainly.com/question/33234150
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