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, 
.
![[asy] dot((5,0)); dot((-5,0)); draw(circle((0,0),5)); dot((3,4)); draw((5,0)--(3,4)); draw((-5,0)--(3,4)); draw((-5,0)--(5,0)); label("A",(-5,0),W); label("B",(3,4),NE); label("C",(5,0),E); [/asy]](http://latex.artofproblemsolving.com/0/6/2/062aa7b165c4c2fe0bd64e4e650d49adbee67d02.png)
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!
