Question on Circles, Finding unknown values of attributes of the circle

Circle is a geometrical figure with basic attributes diameter, radius, circumference, arc, chord etc. Below is a Geometry construction for the circle and to find unknown attributes value.

Topic : Circle and its attributes

Tangent is a straight line which just touches the circumference of the circle at a single point.

Question : Given VQ is tangent to the circle O at Q, QS is a diameter of the circle O, arc PQ = 115⁰, angle RPS = 36⁰

Find a) angle R ; b) angle S ; c) arc SR ; d) arc QR ; e) angle QPR ; f) angle QPS ; g) angle QTP











Solution :











Given : mPQ = 115⁰
‹RPS = 36⁰

a) ‹R
Since QS is the diameter,
So ‹QRS = 90⁰
(as angle in a semi circle is a right angle)

b) ‹S
As mPQ = 115⁰
So ‹PSQ = 1/2 * (115⁰) = 57.5⁰
(angle subtented by an arc at any point on the circle is half of the angle subtented by it at the center)

c) mSR
given ‹SPR = 36⁰
So mSR = 2(36) = 72⁰

d) mQR
now mQS = 180⁰
mQR = mQS - mSR
mQR = 180⁰ - 72⁰
mQR = 108⁰

e) ‹QPR
mQR = 108⁰
So ‹QPR = 1/2*(108) = 54⁰

f) ‹QPS
‹QPS = 90⁰(as QS is the diameter)

g) ‹QTP
mQP = 115⁰
so ‹QSP = 1/2 *(115⁰) = 57.5⁰
now in triangle PTS
‹QTP = 180⁰ - (‹TPS + ‹PST)
‹QTP = 180⁰ - (36⁰ + 57.5⁰)
‹QTP = 180⁰ - 93.5⁰
‹QTP = 86.5⁰

Well these are the bits of circle, to get more examples and get practiced, contact geometry help.