Circle Geometry Example Question (worked proof)

[luisisaurio]

<COE=2x but arc CE=ED there for <EOD=2x. OE=OD cause they are radius, therefore <DEO=90-x. So <EQR=90 which is enough for QRPO to be cyclic

[dawnelysabeth]

for the second question, , is there a way to prove that the sum of the remaining two angles of the cyclic quadrilateral is equivalent to 180°? also, is it possible to find their values with the information given?

[ientonghien]

(i) arcED= arc CE
=> angle CDE= angle DCE (Inscribed angle) (1)
have: EOD is central angle (2)
from (1), (2),
we have angle EOD=2x
(ii) from (i) we have : ECD is isosceles triangle(3)
also have: E is the midpoint of arc CD(4)
from (3), (4) we have EQ is perpendicular to CD
=> angle OQR=angle RPO= 90
conclude:
OPRQ is a cyclic quadrilateral.

[menachemsalomon]

If all we need to prove is that <OQR is 90°, wouldn't it be enough to say that E is the midpoint of AD, then the radius to E bisects the chord AD, and a radius bisects a chord at a right angle? (Although Eddie may be going about proving just that, instead of accepting it as already proven.)

What is a cyclic quadrilateral, BTW? (I'll just look it up.)

[williamnguyen]

Please do more questions like these I really struggle with this type of questions

[ConTejasMusic]

Could you also have used the supplementary angles theorem for part ii?

[ivanhenrickdepedro]

More of these videos!It is really interesting.📖📚📒🔭📓📒

[kuchoomahva]

O my dear, you are really great. Thanks a lot.

[zolt]

I had no idea what he was talking about.

[loganv]

Correct me, but:
arcCE = arcDE
therefore line-segmentCE = line-segmentDE
therefore triangleCDE is isosceles
therefore angleCDE = angleDCE = x degrees
therefore angleEOD = 2x degrees because angle at center = twice angle at circumference

[particleonazock]

Yep, they throw a whole lot of information at you. Especially at James Ruse where the Year 10 exams are like pi^2 times harder than the average school's.

[kuchoomahva]

In India we have to study such questions a lot.

[claudiofelleca]

Hi, what kind of software are you using?

[ishworbasyal]

I dont think this is high school exam. I mean that question was too easy. It just included properties of isosceles triangle, some axioms on circle and manipulation of angles.
I thought you taught to high school students'.