A circle has a radius of 90
centimeters. A point lies on the circle at a
distance of three 𝑥 minus three centimeters from the center. Which of the following is true? (A) 𝑥 equals 31, (𝐵) 𝑥 is
greater than 31, or (C) 𝑥 is less than 31.
In order to answer this question,
we’re going to think about what we know about positions of points and straight lines
with respect to circles. Suppose we have a circle with
center 𝑚. Suppose then we have a point 𝐴
that lies on the circumference of that circle. The radius is the straight-line
segment that joins point 𝑚 to point 𝐴. In the case of our circle center 𝑚
then, 𝑚𝐴 is 90 centimeters. Now, of course, a key property here
is that any line that joins a point at the center of the circle to a point on the
circumference is the radius of that circle. So say we had point 𝐵 somewhere
else on the circle. 𝑚𝐵 is also 90 centimeters.
We’re told that a point that lies
on the circle lies at a distance of three 𝑥 minus three centimeters from the
center. So we can in fact form an equation
using all of the information we’ve looked at so far. If a point lies on the circle at a
distance of three 𝑥 minus three from the center, then we can say that the radius in
this case is three 𝑥 minus three centimeters. But we’ve demonstrated that the
radius is 90. So our equation is three 𝑥 minus
three equals 90. Our job is to solve this for
𝑥. So we’re going to add three to both
sides, giving us three 𝑥 equals 93.
Finally, we solve for 𝑥 by
dividing through by three. And that gives us 𝑥 equals 31. Comparing this to the options in
our question, and we see that the answer is (A). Given the information about our
circle, the statement that is true is (A), 𝑥 equals 31.