Question

See picture attached
how to derive 2theta in the middle?
The only information given in the question is the 2 theta on the top

Answer 1

@cmkc109 are you here?

Answer 2

yes

Answer 3

you want to find \(2\theta\) in terms of what?

Answer 4

the 2 theta i circled is not given in the question
so i want to know how to get that 2 theta

Answer 5

my teacher said is some kind of rule of triangle..which i dont remember

Answer 6

Oh that's easy!!, Let's derive that

Answer 7

really haha, thanks!

Answer 8

Ok, just check the relation between the sides of triangle AOB and triangle AOD?

Answer 9

they are the same..? since the sides are the same

Answer 10

\[\text{AR=AR}\]
\[\text{AD=AB}\]
\[\text{OD=OB}\]
so the triangle AOB and AOD are congruent using SSS theorem

Answer 11

let me refresh the browser..the equations are not showing

Answer 12

Yeah refresh, if you still have viewing problems, do tell me. I'll post again

Answer 13

i really do still see math processing error

Answer 14

AR=AR
AD=AB
OD=OB
so the triangle AOB and AOD are congruent using SSS theorem

Answer 15

yes i get this part

Answer 16

also AR=OD=OR the radius of the circle.
Now in triangle AOD
AO=OD
now we know that angles opposite to equal sides are equal
so
angle OAD=angle ODA= theta
Do you get this?

Answer 17

do u mean AO = OD = OR? AR will be 2 lines isnt it?

Answer 18

Sorry
AO=OD=OB= R

Answer 19

how do you know AO = OD ? cant they be different lengths?

Answer 20

Any line from the center of circle to its circumference is radius

Answer 21

ohhh i get itt

Answer 22

Similarly in triangle AOB, we can prove
angle ABO= angle OAB= theta

Answer 23

yep, that's because they have the same side lengths right?

Answer 24

Yeah !!

Answer 25

What's the sum of angles in a triangle?

Answer 26

180

Answer 27

Great can you find angle AOD in terms of theta using this 180 degree property

Answer 28

is it..180-3theta
but u dont know the value of theta, so there isnt a exact number..

Answer 29

Yeah we want in terms of theta, but how did you get 3 theta?
Check again!!!

Answer 30

oh sorry..it's 2 theta

Answer 31

Cool!!! What's angle AOD in terms of theta?

Answer 32

90?

Answer 33

wait nvm lol

Answer 34

Check again

Answer 35

isnt it just 180-2theta..? am i suppose to rearrange something?

Answer 36

Nope, Let's work with this only

Answer 37

Ok Can you find angle DOB now?

Answer 38

isnt it the same as AOD?

Answer 39

actually..the sides are only the same in the 2upper triangles not the bottom one?

Answer 40

Nope , do you agree with this?
angle AOB+ angle AOD+ angle DOB=360?

Answer 41

the circle is 360

so DOB is 360 - (180 2theta) x2
the 360 cancels out and left with 2theta..

Answer 42

with 2theta?!!!
Check again:D

Answer 43

no, it's 4theta, cuz 2 theta x2
and then divide it by 2, and get that? is it right?

Answer 44

Oh, we didn't need to go this way.

Answer 45

another problem now...cuz dont u do the multiplication before subtraction..

Answer 46

What's angle AOB?

Answer 47

180-2theta
angle AOB = angle AOD

Answer 48

angle BOC +angle AOB= 180
any doubt with this?

Answer 49

why... (srry why taking so much of ur time..:/

Answer 50

Oh no worries, I'm here to help. Wanna make sure you understand everything:D

Answer 51

BOC and AOB are supplementary angles!!

Answer 52

ohh, i know it is 180..lol..looked at the wrong angle

Answer 53

Great:) find BOC now?

Answer 54

180 - (180 - 2theta)
= 180 - 180 + 2 theta
= 2theta?

Answer 55

Yeah:)
So you have your answer:)
Great work

Answer 56

thanks so much :) i get it !