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Question Video: Finding the Radius of a Circle Using the Perpendicular Bisector of a Chord Mathematics • 11th Grade

The chord 𝐴𝐡 of a circle with center 𝑀 has length 28. Suppose the radius 𝑀𝐷 meets 𝐴𝐡 in a right angle at the point 𝐢, and that 𝐢𝐷 = 9.8. Find the radius of the circle.

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Video Transcript

The chord 𝐴𝐡 of a circle with center 𝑀 has length 28. Suppose the radius 𝑀𝐷 meets 𝐴𝐡 in a right angle at the point 𝐢 and that 𝐢𝐷 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 𝐴𝐡, our radius 𝑀𝐷, and the point of intersection 𝐢. 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 𝐴𝐡 in half. Thus, the segment 𝐴𝐢 has half the length of the segment 𝐴𝐡, 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 π‘Ÿ. We can see from the diagram that the length of 𝑀𝐢 is the length of 𝑀𝐷 minus the length of 𝐢𝐷. 𝑀𝐷 is a radius and therefore has length π‘Ÿ. And we’re told that the length of 𝐢𝐷 is 9.8. Therefore, 𝑀𝐢 has length π‘Ÿ minus 9.8. Since 𝐴𝑀𝐢 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.

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