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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. AOB = CO’D (it is given that arcs are congruent) Hence
by SAS Postulate on congruence AOB CO’D 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 = 2r, Area of the circle = r2, Where is a constant whose
approximate value we use for our calculations is 22/7 (3.1428). If (where is in degrees) is
the angle at center (COD) formed by the arc CSD then 1. Length
of the arc CSD = (/180) *r 2. Area
of the sector CSDO (shaded
portion in the adjoining figure) = (/360) *r2 = (/180) *(r*r)/2 = {(/180) *r}*(r/2) = Length of the arc*(radius/2) Note: radians = 1800
and x0 = (x*)/180 radians |
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Let AOB = in the adjoining figure with AB as chord We note that Area of triangle ABO = (1/2)*base*height = (1/2)*BO*AM = (1/2) *r*rsin= (1/2) r2*sin (AM = rsin : Refer to section 7.1 for definition of sin of an
angle) From the figure we notice that Area of Sector ASBO = Area of triangle ABO + Area of
segment ASB Area of segment ASB = Area of Sector ASBO - Area of
triangle ABO = (/360) *r2 - (1/2) r2*sin = r2 {(*/360) - (sin/2)} Note: For all the above calculations must be in degrees. |
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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 |