# Difference between revisions of "Thales' theorem"

Line 22: | Line 22: | ||

'''Problems''' | '''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. | ||

## 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!*