# Difference between revisions of "Thales' theorem"

(added one more problem) |
|||

Line 1: | Line 1: | ||

− | 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>. |

Line 17: | Line 17: | ||

− | 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. | ||

Line 36: | Line 36: | ||

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

+ | [[Thales]] |

## Latest revision as of 16:52, 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