need help with circle geometry Qs !!! (1 Viewer)

[supermike]

hi guys i need ur brains on these :

find angles alpha and beta. Reasons must be indicated. hint: this is a question from a unit on tangent and radius drawn to point of contact and tagent from external point.(circle geometry)

prove that triangle PAT is similar to triangle QBT

any help would be much appreciated!!!

 

[44Ronin]

Pi

 

[Drongoski]

d) alpha = 100 deg; beta = 30 deg ??

Let O be centre of circle and pt @ 140 deg be E; reflex angle COB = 2 x 140 deg

.: angl BOC = 360 - 280 = 80 deg ==> Angl BCO = angl CBO = 50 deg

.: angl BCE = 90deg - 50 - 10 deg = 30 deg (angl OCA = 90 deg)

.: alpha = 180 - 2 x angl BCA = 180 - 2 x 40 = 100 deg

.: beta = angl BCE = 30 deg (angl in alternate segment)





(a) In trgls PAT & QBT:

angl PTA = angl QTB (vert opp angls)

angl APT = ang BQT ( angls in a semi=circle = 90 deg)

.: 2 triangles are equi-angular and .: similar

 

[Air Jordan]

PTA = BTQ (vertically opposite angles are equal)
APT = BQT (equal right angles... not really sure how to say it)
PAT = TBQ (remaining angles equal?)

Therefore triangle PAT is similar to triangle QTB (Equiangular)

 

[supermike]

d) alpha = 100 deg; beta = 30 deg ??

you are right!!!
thanks for the help so far guys!! i just realised , with ur help, second question is ridiculously easy and i am already bearing the shame i deserve.
however i would like to see a more reason based 3-unit approach to the first question if u dont mind

ths

 

[Drongoski]

you are right!!!
thanks for the help so far guys!! i just realised , with ur help, second question is ridiculously easy and i am already bearing the shame i deserve.
however i would like to see a more reason based 3-unit approach to the first question if u dont mind

ths

Sorry I had no time to go into detail.

1) angle @ centre of circle = 2 x angle @ circumference; this one applies to the 'reflex angle' (more than 180 deg one)

2) OB and OC the 2 radii make a right angle with each of the 2 tangents AB and AC.

3) make sure u mark the centre O and the point E and join OB, OC and BC.

Without a diagram hard for me to set out proof as I'd have liked.

 

[supermike]

Sorry I had no time to go into detail.

1) angle @ centre of circle = 2 x angle @ circumference; this one applies to the 'reflex angle' (more than 180 deg one)

2) OB and OC the 2 radii make a right angle with each of the 2 tangents AB and AC.

3) make sure u mark the centre O and the point E and join OB, OC and BC.

Without a diagram hard for me to set out proof as I'd have liked.

ths dude U R THE BOMB!!!