Circle Geometry Question...? (1 Viewer)

[pinupgirl]

Just a question, I found in the Yr 12 Cambridge book ...
It looks simple but I'm stuck o.o

"Prove that if two chords of a circle bisect each other, then they are both diameters."

Help please?
Thanks!

 

[SpiralFlex]

I drew a pretty diagram for you.




Let , also let the intersection of the chords be

(Angles at the circumference subtended by the same arc.)

Similarly,



Hence, (Corresponding sides of congruent triangles.)


Let

(Angle at the circumference subtended by the same arc.)

Similarly,

Now,

(Corresponding sides in congruent triangles.)



Now, quadrilateral ,

(Opposite interior angles in a cyclic quadrilateral are supplementary.)

Now in and ,

(As proven.)

(As proven.)

is common.

Hence,

,

Now,

(Corresponding angles in congruent triangles.)

Back to our equation of,







Hence is the diameter, angle at the semi circle is 90.

 

[Drongoski]

In geometry there are often many ways to skin a cat. How you do your proof also depends on prior results (theorems) you may assume. So another approach is (using SpiralFlex's diagram):


1: Since diagonals AC & BD bisect each other ABCD is a //gram.

2: Since AC = BD this //gram is a rectangle

3: .: angles ADC and BAD (being angles of the rectangle) are right angles.

4: .: AC and BD are diameters of the circle (since ADC & BAD are angles in a semi-circle)


QED

 

[funnytomato]

AC and BD bisects each other---> ABCD is a parallelogram--->opposite angles are equal, so let ∠ABC and ∠ADC both be x degrees

but ABCD is also a cylic quad, so opposite angles are supplementary
x+x=180°， x=90° ---> AC is a diameter

similarly BD is also a diameter

 

[Hermes1]

AC and BD bisects each other---> ABCD is a parallelogram--->opposite angles are equal, so let ∠ABC and ∠ADC both be x degrees

but ABCD is also a cylic quad, so opposite angles are supplementary
x+x=180°， x=90° ---> AC is a diameter

similarly BD is also a diameter

nice.

 

[pinupgirl]

Thanks everyone so much!