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

The chord π΄π΅ of a circle with center π has length 28. Suppose the radius ππ· meets π΄π΅ in a right angle at the point πΆ, and that πΆπ· = 9.8. Find the radius of the circle.

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### Video Transcript

The chord π΄π΅ of a circle with center π has length 28. Suppose the radius ππ· meets π΄π΅ in a right angle at the point πΆ and that πΆπ· 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 π, our chord π΄π΅, our radius ππ·, and the point of intersection πΆ. 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 circleβs center, and so our radius ππ· cuts the chord π΄π΅ in half. Thus, the segment π΄πΆ has half the length of the segment π΄π΅, that is, 28 divided by two, which equals 14.

The next thing we know is that π΄π is also a radius. Letβs call its length π. We can see from the diagram that the length of ππΆ is the length of ππ· minus the length of πΆπ·. ππ· is a radius and therefore has length π. And weβre told that the length of πΆπ· is 9.8. Therefore, ππΆ has length π minus 9.8. Since π΄ππΆ 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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