# Difference between revisions of "Thales' theorem"

Line 19: | Line 19: | ||

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 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 problems; however, the name of the theorem is not well-known. Thus, you may cite the "universal fact" that <ABC = | + | 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</math> in proofs without specifically referring to Thales. |

==Problems== | ==Problems== |

## Revision as of 00:41, 5 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