Question

In the figure to the right, lines that appear to be tangent are tangent, and O is the center of the circle.What is the value of x?69degrees

Answer 1

To solve this problem, we will use the following fact about circles. Given the circle

whenever we are given two tangent lines, the angle formed between then (in our picture angle theta or angle ABC) is related to the intercepted arcs as follows

$\theta=\frac{1}{2}(y-x)$

where y and x are measured in degrees.

For our particular problem, we have the following

So, using the fact that we introduced at the beginning, we have the following equation

$69=\frac{1}{2}(y-s)$

However, note that since arcs y and s form the whole circle, this means that their measure (in degrees) should add up to the measure of the whole circle (360°). So we have the following equation

$s+y=360$

So, we can subtract s on both sides to get

$y=360-s$

Now, we replace this value of y in our first equation, so we get

$69=\frac{1}{2}((360-s)-s)=\frac{1}{2}(360-2s)$

Note that arc s is the one described by the central angle x. So, this means that

$x=s$

So if we replace this in our equation, we get

$69=\frac{1}{2}(360-2x)$

If we multiply both sides by 2, we get

$360-2x=69\cdot2=138$

Now, we can add 2x on both sides, so we get

$360=138+2x$

Then, we subtract 138 on both sides, so we get

$2x=360-138=222$

Now, we divide both sides by 2. So we get

$x=\frac{222}{2}=111$

So the value of x is 111°