Difference between revisions of "Thales' theorem"
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− | Thales' Theorem states that if there are three points on a circle, <math>A,B,C</math> with <math>AC</math> being a diameter, <math>\angle ABC=90^{\circ}</math>. | + | [[Thales]]' Theorem states that if there are three points on a circle, <math>A, B, C</math> with <math>AC</math> being a diameter, <math>\angle ABC=90^{\circ}</math>. |
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− | This is easily proven by considering that the intercepted arc is a semicircle | + | This is easily proven by considering that the intercepted arc is a semicircle or 180°. Thus, the intercepted angle is 180°/2 = 90°. |
− | This theorem has many uses in geometry because it helps introduce right angles into | + | This theorem has many uses in geometry because it helps introduce right angles into problems; however, the name of the theorem is not well-known. Thus, you may cite the "universal fact" that <ABC = 90° in proofs without specifically referring to Thales. |
− | + | ==Problems== | |
1. Prove that the converse of the theorem holds: if <math>\angle ABC = 90^{\circ}</math>, <math>AC</math> is a diameter. | 1. Prove that the converse of the theorem holds: if <math>\angle ABC = 90^{\circ}</math>, <math>AC</math> is a diameter. | ||
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''Please add more problems!'' | ''Please add more problems!'' | ||
+ | [[Thales]] |
Revision as of 15:52, 25 November 2019
Thales' Theorem states that if there are three points on a circle, with
being a diameter,
.
This is easily proven by considering that the intercepted arc is a semicircle or 180°. Thus, the intercepted angle is 180°/2 = 90°.
This theorem has many uses in geometry because it helps introduce right angles into problems; however, the name of the theorem is not well-known. Thus, you may cite the "universal fact" that <ABC = 90° in proofs without specifically referring to Thales.
Problems
1. Prove that the converse of the theorem holds: if ,
is a diameter.
2. Prove that if rectangle is inscribed in a circle, then
and
are diameters. (Thus,
.)
3. is a diameter to circle O with radius 5. If B is on O and
, then find
.
4. Prove that in a right triangle with AD the median to the hypotenuse, .
5. is a diameter to circle O, B is on O, and D is on the extension of segment
such that
is tangent to O. If the radius of O is 5 and
, find
.
6. In a triangle ,
is the median to the side
(
is the midpoint). If
, then prove that
without using Thales' theorem. If you have a general understanding of how the theorem works and its proof you can manipulate it into the solution.
Please add more problems! Thales