Difference between revisions of "Thales' theorem"

 
(One intermediate revision by one other user not shown)
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, or 180°. Thus, the intercepted angle is 180°/2 = 90°.
+
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.
+
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'''
+
==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 32: Line 32:
  
 
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>.
 +
 +
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 15:52, 25 November 2019

Thales' Theorem states that if there are three points on a circle, $A, B, C$ with $AC$ being a diameter, $\angle ABC=90^{\circ}$.


[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]


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 $\angle ABC = 90^{\circ}$, $AC$ is a diameter.

2. Prove that if rectangle $ABCD$ is inscribed in a circle, then $AC$ and $BD$ are diameters. (Thus, $AC = BD$.)

3. $AC$ is a diameter to circle O with radius 5. If B is on O and $AB = 6$, then find $BC$.

4. Prove that in a right triangle with AD the median to the hypotenuse, $AD = BD = CD$.

5. $AC$ is a diameter to circle O, B is on O, and D is on the extension of segment $BC$ such that $AD$ is tangent to O. If the radius of O is 5 and $AD = 24$, find $AB$.

6. In a triangle $ABC$, $CD$ is the median to the side $AB$($D$ is the midpoint). If $CD=BD$, then prove that $\angle C=90^\circ$ 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

Invalid username
Login to AoPS