6.12 Circles - Part 3:
6.12.1:
Arcs of a circle
Two arcs of two different circles having same radii are said to
be ‘congruent’ if their central angles
are same.
Arc ASB = Arc CTD if AOB =
CO’D
6.12.1 Theorem 1: If two arcs are congruent then their chords are equal
To prove: AB=CD
Proof:
1. OA = O’C, 2.
Hence
by SAS Postulate on congruence Hence AB = CD |
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6.12.1 Theorem 2: If two chords of circles having same radii are same,
then their arcs are congruent.
Note: This is converse of the previous theorem.
Use SSS postulate to show that AOB =
CO’D
6.12.1:
Areas of sectors/segments of circle
If ‘r’ is the radius of a
circle, we know that the circumference and area of the circle are given by Circumference of the circle = 2 Area
of the circle = Where If 1. Length of the arc CSD = ( 2. Area of the sector CSDO (shaded portion in the adjoining figure) =
( = ( = Length of the arc*(radius/2) Note: |
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Let We note that Area of triangle ABO =
(1/2)*base*height = (1/2)*BO*AM = (1/2) *r*rsin (AM = rsin From the figure we notice that Area of Sector ASBO = Area of
triangle ABO + Area of segment ASB
= ( = r2 {( Note: For all the above
calculations |
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6.12 Summary of learning
No |
Points to remember |
1 |
Congruency
of arcs |
2 |
Formula
for length of an arc, area of an arc, Area of a segment |