Circle Geometry: Line Passing Through Centre of Circle (1 Viewer)

[TrueHappiness]

AB and CD are two parallel chords of a circle. CA and DB meet, when produced, at X. AD and BC intersect at Y

There are three parts to this question, two of which are already solved and i will post results:

a)YA= YB
b)XA= XB

How must i prove that XY will pass through center?

And just generally, how is this achieved without using co-ordinate geometry, i.e. proving a certain line passes through center.

 

[random-1006]

AB and CD are two parallel chords of a circle. CA and DB meet, when produced, at X. AD and BC intersect at Y

There are three parts to this question, two of which are already solved and i will post results:

a)YA= YB
b)XA= XB

How must i prove that XY will pass through center?

And just generally, how is this achieved without using co-ordinate geometry, i.e. proving a certain line passes through center.

ok, got it

now from parts a and b, be know that AYBX is a kite ( adjacent sides equal), now whats special about a kite, the diagonals meet at right angles.

so we have XY is perpendicular to AB

now there is a theorm, a line from the centre to a chord, is perpendicular to the chord ( and bisects the chord ) , you can prove that it bisects the chord ( however, im not 100% sure if thats needed for this question, i dont know my theorms word for word as i havent done much 2/3 unit in the past yr other than the questions i see on here) by simple congruent triangles ( same as you would have done for part a) .

HOWEVER, there is no simple method for determining a line go's through the centre , if i gave you a circle and it had a line going between two points and that line subtended a right angle at the edge, you would know that the line that was drawn on the circle was a diameter ( ie goes through the centre) ( because angle at centre= twice angle at edge)

You just need to know the theorms, and that you can apply them in reverse.