# Difference between revisions of "Thales' theorem"

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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>. | + | [[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 | + | This is proven by considering that the intercepted arc is a semicircle and has measure <math>180^{\circ}</math>. Thus, the intercepted angle has degree measure <math>\frac{180}{2} = 90</math>. |

− | 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 <math>\angle ABC = 90^{\circ}</math> in proofs without specifically referring to Thales. |

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+ | ==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. | ||

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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!'' | ||

+ | [[Thales]] |

## Latest revision as of 23:42, 4 June 2021

Thales' Theorem states that if there are three points on a circle, with being a diameter, .

This is proven by considering that the intercepted arc is a semicircle and has measure . Thus, the intercepted angle has degree measure .

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 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