Video Transcript
The chord 𝐴𝐵 of a circle with
center 𝑀 has length 28. Suppose the radius 𝑀𝐷 meets 𝐴𝐵
in a right angle at the point 𝐶 and that 𝐶𝐷 has length 9.8. Find the radius of the circle.
The first thing to do is to draw a
sketch. Here we can see our circle with
center 𝑀, our chord 𝐴𝐵, our radius 𝑀𝐷, and the point of intersection 𝐶. A good approach for questions like
this is to write down everything we know and then try to form an equation. Recall first that the chord
bisector theorem tells us that any straight line passing through the center of a
circle and intersecting a chord in a right angle must cut that chord in half. In particular, any radius is a
straight line passing though the circle’s center, and so our radius 𝑀𝐷 cuts the
chord 𝐴𝐵 in half. Thus, the segment 𝐴𝐶 has half the
length of the segment 𝐴𝐵, that is, 28 divided by two, which equals 14.
The next thing we know is that 𝐴𝑀
is also a radius. Let’s call its length 𝑟. We can see from the diagram that
the length of 𝑀𝐶 is the length of 𝑀𝐷 minus the length of 𝐶𝐷. 𝑀𝐷 is a radius and therefore has
length 𝑟. And we’re told that the length of
𝐶𝐷 is 9.8. Therefore, 𝑀𝐶 has length 𝑟 minus
9.8. Since 𝐴𝑀𝐶 is a right triangle,
we can apply the Pythagorean theorem. We substitute in our values,
multiply out, and simplify. The radius is 292.04 divided by
19.6, which is 14.9.
