Answer:
x = 9
Step-by-step explanation:
The total angle in a circle measures to be 360 degrees
Based on the given information, this is the measure of the smaller arc:
360 - (23x - 3)
Apply theorems to find an equation.
Intersecting Secant and Tangent Theorem: When a line is tangent with a circle at the point of tangency, the angles on the circle's interior such as 78 degrees will be half the amount of the angles on the exterior (intercepted arcs).
Therefore our equation will be the following, let's solve it all the way through:
78 = 1/2 (360 - (23x - 3))
78 = 1/2 (360 - 23x + 3)
78 = 1/2 (363 - 23x)
78 = 363/2 - 23/2x
*2 *2 *2 <--- Multiply everything by 2 to remove the fractions.
156 = 363 - 23x
+23x +23x
------------------------
363 = 156 + 23x
-156 -156
-----------------------
207 = 23x <--- Divide both sides of the equation by 23.
x = 9
Therefore, x = 9 is our most simplified, final answer.
