Properties of Circle 

 

 

 

Contents

 

Symmetrical Properties 

 

Angle Properties

 

Cyclic Quadrilaterals

 

Problems on angle properties

 

Tangents from External Point

 

Alternate Angle Segment

 

Quiz

 

 

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Alternate Segment Theorem

The alternate segment theorem states that an angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. Thus, ÐPTB = ÐPQT.

Proof:

      ÐSPT = 90° (rt. Ð in a semicircle)

Ðx + Ðy = 90° ( Ð sum of ∆)
Ðy + Ðz = 90° ( tan rad.)
            Ðx = Ðz
     ÐPTB = ÐPST
     ÐPST = ÐPQT ( Ðs in same segment)
     ÐPTB = ÐPQT

Example 1

In the figure shown, PA and PB are tangents to the circle. ABP = 60° and BAC = 40°. Find the value of x and y.

x° = 60° (Ð in alt. segment)

ÐPAB = 40° (base Ð of isos. ∆PAB)
   y° = 180° - 60° - 40° (adj. Ðs on str. l)
       = 80° 
 
x is 60 and y is 80.

Example 2
 
Calculate the value of x and y.
 
ÐBAC = 90° (rt.Ð in semicircle)
ÐOBA = x° (Ð in alt. segment)
     x° = 180° - 90° - 55° (Ð sum of ∆)
         = 35°
     y° = 180° - 90° - 35° - 35° (Ð sum of ∆)
         = 20°