Circle Geometry: Shortest Chord (1 Viewer)

[Lukybear]

AB, CD and XY are chords in a circle with centre O. XY cuts AB and CD in L and M, which are the midpoints of AB and CD. Prove that XY is greater than either AB or CD.

I hav no idea how to approach this problem

 

[Rezen]

Interesting question

construction: let the midpoint of XY by P. let the radius be x units long.

Proof: To prove XY>AB and CD is the same as proving OP < OM and OL. as lines from origin to midpoints of chords are perpendicular to the chord and the closer the chord is to the origin the bigger it is.

now, XM.MY=AM.MB (intercepts on intersecting chords)
but M is midpoint of AB. therefore XM.MY=AM^2 .....(1)

similarly for other chord, XL.LY=CL^2 ....(2)

In triangle AMO, angleAMO=90 (midpoint perp to origin)

therefore, AM^2+OM^2=x^2 (pythagoras, radius is x units long)

and OM=sqrt(x^2-AM^2)=sqrt(x^2-XM.YM), from (1)

similarly for triangle OCL and from (2)
OL=sqrt(x^2-XL.LY)

similarly for triangle XOP,
OP=sqrt(x^2-XP^2)

we want to prove OP < OM and OL

from these equations we need to show that x^2-XP^2< x^2-XM.MY and x^2-XL.LY

or that XP^2>XM.MY and XL.LY

ie, that the product of two intersections of a segment is greatest when the intersection is taken from the midpoint. (not sure how to word it here, but hopefully its understandable what i mean.)

proof of this: let the line segment be E units long. take one section as y units, then the other will be (E-Y). let the product be F. therefore F=y(E-y)=Ey-y^2.
differentiating, F'=E-2y. F'=0 when y=E/2. ie product of the two sections is greatest when they are halved.

applying this to circle question, therefore XP^2 > XL.LY and XM.MY, which implies OP < OM and OL

and therefore XY > AB and CD


i know this is a pretty dodgey proof but i dont really want to spend too much time writing this. Also if anyone can think of a shorter proof, let me know.

 

[angrygeorge]

have you tried the triangle theorm (sum of 2 sides greater than third side?)

 

[shaon0]

have you tried the triangle theorm (sum of 2 sides greater than third side?)

Triangle Inequality

 

[hscishard]

Triangle Inequality

What the hell is that?

 

[shaon0]

What the hell is that?

 

[random-1006]

actually say it it rather some writting some bullshit expression no one will understand. ( they probably will, but will take them a while)

"The sum of two sides of a triangle must be shorter than ( or equal) the length of the other side"

 

[hscishard]

Is this in a textbook?

 

[shaon0]

actually say it it rather some writting some bullshit expression no one will understand. ( they probably will, but will take them a while)

"The sum of two sides of a triangle must be shorter than ( or equal) the length of the other side"

It was already stated above dumbshit, why would i restate it.

Yes, it's in Cambridge under harder 3u.

 

[hscishard]

I'm guessing 4unit?

 

[shaon0]

I'm guessing 4unit?

Oh sorry, idk if 3u has to know triangle inequality. No harm in knowing about it but they'll never ask you to use it in the exam (even implicitly).