Difference between revisions of "Thales' theorem"
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5. <math>AC</math> is a diameter to circle O, B is on O, and D is on the extension of segment <math>BC</math> such that <math>AD</math> is tangent to O. If the radius of O is 5 and <math>AD = 24</math>, find <math>AB</math>. | 5. <math>AC</math> is a diameter to circle O, B is on O, and D is on the extension of segment <math>BC</math> such that <math>AD</math> is tangent to O. If the radius of O is 5 and <math>AD = 24</math>, find <math>AB</math>. | ||
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+ | 6. In a triangle <math>ABC</math>, <math>CD</math> is the median to the side <math>AB</math>(<math>D</math> is the midpoint). If <math>CD=BD</math>, then prove that <math>\angle C=90^\circ</math> 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!'' | ''Please add more problems!'' |
Revision as of 23:17, 29 November 2016
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 a problems; however, the name of the theorem is not well-known. Thus, you may cite the "universal fact" that <ABC = 90° in a proof 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!