I have a question (given by a teacher) that looks really easy but then when I thought about it, couldn't find a way to find the answer. It is a proof question relating to diameters:
Prove that any two diameters are congruent, and that each diameter is twice as long as each radius.
It seems like common sense, but I can't think of a way to prove it.
Besides this, I also have another question (that I stumbled upon i my reference book) about circles that I could not figure out:
If one chord is a perpendicular bisector of a different chord, then the first chord is a diameter
I think there must be a property relating the center of a circle and the perpendicular bisector of any chord, but I am not sure where to start with this problem.
Any help or hints would be very nice 🙂
Thank you.
Best Answer
Hint for first part: Consider two diameters on a circle, of length $a$ and $b$. Since a diameter is a chord of maximum length, diameter $b$ is greater than or equal to all other possible chord lengths, including $a$. Similarly diameter $a$ is greater than or equal tochord $b$. But $a\leq b$ and $b\leq a$ together imply $a=b$. Hence, any two diameters have the same length, hence any two diameters are congruent.
To show that this maximum chord length is $2r$, you need to show that chords which go through the center are longer than chords which don't.