Point C is the center of the circle. The measure of angle ACB is 15x + 20. Arc AB measures 20x – 10. Find the measure of arc AB.
To find the measure of arc AB, we need to connect it to the angle ACB to create a relationship between them.
In a circle, an inscribed angle is equal to half the measure of the intercepted arc. This means that the measure of angle ACB is equal to half the measure of arc AB.
Given that the measure of angle ACB is 15x + 20 and we want to find the measure of arc AB, which is 20x – 10, we can set up the equation:
15x + 20 = (1/2)(20x – 10)
To solve this equation, we first distribute the 1/2 to the terms inside the parentheses:
15x + 20 = (1/2)(20x) – (1/2)(10)
Simplifying further, we get:
15x + 20 = 10x – 5
Now, let's isolate the variable x by moving all x terms to one side and all constant terms to the other side:
15x - 10x = -5 - 20
Combining like terms, we have:
5x = -25
Finally, solving for x by dividing both sides of the equation by 5:
x = -25/5
x = -5
Now that we have found the value of x, we can substitute it back into the expression for arc AB:
Arc AB = 20x – 10 = 20(-5) – 10 = -100 – 10 = -110
Therefore, the measure of arc AB is -110 units.