proof question geometry

[solar moss]

not sure what theorem needs to be used to prove it

[lusty ether]

I have an easy way out for this

And they have not mentioned anything abt AD

Consider it as diameter

[solar moss]

consider ad a diameter?

[lusty ether]

Yepp

[sour latch]

Why

[lusty ether]

Just to make the proof

Easy

For convenience

[solar moss]

alr

[sour latch]

Will it always be diameter?

[solar moss]

no

[sour latch]

Then! Why assume?

[lusty ether]

Ok so the angle circumscribed by the semicircle is 90°

That's the property

So if u consider the figure

[solar moss]

yeah i know what u mean

[lusty ether]

Prove ABD and ACD triangles congruent by hypotenuse side test

And u get ur answer

[solar moss]

but i dont think that proves it since that assumes that if AD is a diameter

heres what i did

[lusty ether]

U want the figure?? Or did u get it

[shy canyon]

its just an angle bisector property

[lusty ether]

Yepp exactly

[shy canyon]

similar angle = similar chord lenght

[solar moss]

ive never seen that theorem before

[lusty ether]

It does actually. It's a direct proof so it's valid

[shy canyon]

there is nothing to discuss, or if u want it more synthetic then angle CAD =OCB and OAB = COB

[lusty ether]

Prove the angles equal

Any 2

That's AA test for similarity

So the sides BD CD have similar lengths

[sour latch]

So you are saying AD is a diameter? If not, you just can't assume it!

[lusty ether]

Nope

In the beginning u say assume AD to be a diameter

As there is no restriction to prove it

[sour latch]

Okay! Then?

[lusty ether]

I mentioned it above

[sour latch]

So you got the proof with the assumption

Is it still true without the assumption?

[solar moss]

idk i proved it with alternate angle theorem

[lusty ether]

Yess it is for any angle bisectors AD on circumference of circle

[lusty ether]

👍 👍

[lusty ether]

U can also prove it

By using parametric form

[solar moss]

ive never heard of that

what is it

[lusty ether]

Consider the points as parameters r cosx, rsinx

r is the radius

x is angle

[solar moss]

that seems like a way overcomplicated way

[lusty ether]

It's easy though just finding 2 equations and adding their squares

But it's higher level

U can close the forum if ur doubt is cleared