Difference between revisions of "Thales' theorem"
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'''Problems''' | '''Problems''' | ||
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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. | ||
Revision as of 18:41, 20 April 2014
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
.
Please add more problems!