On this site, we are not able to include diagrams, so i can only describe what to do. Additionally, I lack a Geometry book so i am unable to give you the exact propositions needed for a geometric proof.
Draw two concentric circles such that one is smaller than the other.
Label the center.
Draw a horizontal chord at the top of the larger circle such that the chord is tangent to the smaller circle.
At the left intersection of the chord and the larger circle, draw a line to the center of the circles.
At the right intersection of the chord and the larger circle, draw a line to the center of the circles.
Starting at the center of the circles, label the triangle A, B, and C with B to the left, and C to the right.
It is clear that segments AB and AC are radii of the larger circle, and are of equal length.
We also know that the angles formed by drawing a line from the center of a circle to the convergences of a chord are equal.
From the center of the two circles, draw a line to the supposed tangent point. Label that point D.
When that step is done, you will have two triangles with two sides of equal lengths and one equal angle.
By Side-Angle-Side, the two triangles are congruent.
Since the triangles are congruent, segment BD = segment DC.
I hope this description is helpful to you.
Hannah M.
Does Angle-Angle-Side also prove that the point of tangency is the midpoint? Since a diameter that bisects a chord (that isn't a diameter) is perendicular to the chord. Thus, making 90 degree angles at point D.
Thanks again!
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05/28/16
Joseph C.
tutor
The proof comes from the fact that segments BD and DC are of equal length.
Thanks for responding. It is always gratifying to hear from students.
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05/28/16
Hannah M.
Ok. Thanks!
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05/28/16
Hannah M.
I appreciate the help you have given to me so far. I appreciate it very much!
I am wondering if you can help me with this question. I have read and reread my textbook and looked all over the web but cannot find an example of how to do this.
I have a circle. The radius is 9. There is a chord drawn across from one side of the circle to the other side. It is perpendicular to the radius. This chord is drawn at the top of the circle. The arc above this chord is 45 degrees. I am to find 1/2 of the length of the chord.
Thank you!
Hannah
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05/30/16
Joseph C.
tutor
Dear Hannah,
Draw a line from the center of the circle to the left point where the chord intersects with the circle. Then, draw a similar line on the right side.
At that point, you will have a triangle. Each of the legs of the triangle will be a radius of the circle, with a length of 9.
The number of degrees in the arc will be equal to the angle formed by the two radii, namely 45o.
Using Side-Angle-Side, you can calculate the length of the third leg, which is the chord. Once you have that length, the number you want will be 1/2 the value.
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05/30/16
Hannah M.
Thank you so much!
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05/30/16
Joseph C.
tutor
Dear Hannah,
You are welcome, and you may send a message to me at any time.
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05/31/16
Hannah M.
05/28/16