# Difference between revisions of "Thales' theorem"

Flamefoxx99 (talk | contribs) (Created page with "Thale's 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> <asy> dot((5,0))...") |
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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>. | |

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− | { | + | 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 <math>\angle ABC = 90^{\circ}</math>, <math>AC</math> is a diameter. | ||

+ | |||

+ | 2. Prove that if rectangle <math>ABCD</math> is inscribed in a circle, then <math>AC</math> and <math>BD</math> are diameters. (Thus, <math>AC = BD</math>.) | ||

+ | |||

+ | 3. <math>AC</math> is a diameter to circle O with radius 5. If B is on O and <math>AB = 6</math>, then find <math>BC</math>. | ||

+ | |||

+ | 4. Prove that in a right triangle with AD the median to the hypotenuse, <math>AD = BD = CD</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>. | ||

+ | |||

+ | ''Please add more problems!'' |

## Revision as of 19:41, 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!*