The discussion revolves around proving that any chord perpendicular to the diameter of a circle is bisected by that diameter. This falls under the subject area of circle geometry and involves properties of chords and diameters.
The discussion is active, with participants engaging in reasoning about triangle congruence and the properties of the circle. There is recognition of the need for careful consideration of assumptions, particularly regarding the congruence criteria being applied.
Participants are navigating the challenge of proving a geometric property without assuming the conclusion, which is a common constraint in mathematical proofs.
What triangles are you forming ?Mr Davis 97 said:Homework Statement
Prove that any chord perpendicular to the diameter of a circle is bisected by the diameter.
Homework Equations
The Attempt at a Solution
I was thinking that maybe I could form two triangles, show that these triangles are congruent, and then conclude that the two lengths of the chord cut by the diameter are equal in length. But I can't seem to prove congruence.
Oh wait... Let X be the intersection of the chord and the diameter. If I form triangles with the radius, then I get that the hypotenuses are equal, but I also get that the segment from X to the center of the circle is the same for both triangles, so they are congruent by SSS (since the other side for both triangles comes from the Pythagorean theorem).SammyS said:What triangles are you forming ?
Yes, the triangles are congruent, but not by SSS. That would require that you assume the thing you are to prove.Mr Davis 97 said:Oh wait... Let X be the intersection of the chord and the diameter. If I form triangles with the radius, then I get that the hypotenuses are equal, but I also get that the segment from X to the center of the circle is the same for both triangles, so they are congruent by SSS (since the other side for both triangles comes from the Pythagorean theorem).