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If two circles are concentric and a chord of the larger circle is tangent to the smaller circle, what is the relationship between the chord and the point of tangency of the smaller circle?

 Mar 25, 2020
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Sorry I don't have an accompanying diagram(I'm not too tech savvy with programs that others are more familiar with like desmos, etc.), but I'll try my best to explain my solution.

We draw a large circle, and within it, another smaller circle that shares the same center(that's the definition of concentric more or less). Then, we draw a chord(a line segment with both endpoints on the circle) across the large circle that touches the smaller circle at exactly one spot. After we've done this, we can then draw a line from the center of the smaller circle to the point of tangency, creating a 90 degree angle. By definition, if a line is tangent to a circle, it forms a 90 degree angle with the radius drawn to that point.

Here's a proof of the following done by Euclid:

https://www.quora.com/Why-does-the-tangent-of-a-circle-make-an-angle-of-90%C2%B0-with-the-radius-drawn-from-the-point-of-contact

 

Next, we notice that we can draw in two radii of the larger circle, both of which intersect an endpoint of the chord. We then form two right triangles that have the same hypotenuse and same leg(which is the radius of the smaller circle). By pythagorean theorem, it stands to reason that if the hypotenuse and leg of a right triangle are congruent to another hypotenuse leg pair, then both right triangles are congruent. For this reason, we know the legs of the two right triangles are equal to each other, meaning that the point of tangency is indeed the midpoint of the chord drawn.

 Mar 25, 2020

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