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this is an amazing question because in
this single question we will be using
four different theorems like cord chord
power theorem perpendicular bis sector
of a chord phases theorem and Pythagoras
Theorem we have a circle now draw this
chord its length is 13 units then draw
this line which is perpendicular to this
chord its length is six units then draw
another line which is perpendicular to
this line its length is 18 units our job
is to find the radius of this circle so
can you solve it okay in order to solve
this problem let us draw a line which
will be parallel to this line like this
now this is 13 and this is 18 units so
what will be the length of this piece
yes you are right it will be 18 -3 or 5
units right now let us draw this line
and complete this chord and let us label
the length of this piece as H now there
is a theorem called the perpendicular
bis sector of a chord theorem which
states that a line drawn from the center
of a circle to a cord which is
perpendicular to the cord will always
BCT it or cut the cord into two equal
halves so let us draw this perpendicular
bis sector line from the center of this
circle using the theorem both of them
will be 13 / 2 now since these two lines
are parallel and this is a right angle
therefore this length will be equal to
this and thus it will be 13 / 2
similarly this will also be
13/2 right now again using the
perpendicular bis sector of a chord
theorem on this chord both these pieces
will be of equal length therefore we get
13 / 2 + H will be equal to this will be
13 /2 + 5 therefore H equal 5 let us
clear things up noise now here starts
the real magic we can now use a theorem
called the chord chord power theorem
imagine two lines crossing inside a
circle making four pieces like this let
us label the sides as a b c and d the
theorem says that if you multiply the
lengths of the two parts of one chord or
a * B then it will always be equal to
the same multiplication for the other
chord or C * D now let us draw a line
like this what will be the length of
this piece of the chord label it as M
look at these four parts of the chord we
can use the chord chord power theorem
here to get this times this or 18 * 5 =
this * this or 6 * m therefore we get M
= 90 over 6 or 15 units so what will be
the length of this cord it will be 15 +
6 or 21 units again let us clear things
up and remove the unnecessary lines
great now let us connect these two
points with each other see clearly I
have made this line pass through the
center of this circle do you know why
here comes another theorem called
thales's theorem which states that if a
triangle is formed inside a circle with
one of its sides as the diameter of the
circle then the angle opposite to the
diameter is always a right angle or 90°
which means it will pass through the
center of this circle now if we label
the radius of this circle as R then this
diameter will be of length 2 * R right
so finally we can use our favorite right
triangle theorem here to get 2 r² =
s + 21 s therefore 4 r² = 169 +
441 or
610 thus r² = 610 / 4 which will be
152.5 and therefore R equals the square
root of this which is approximately
12.35 units and that's it this is our
final answer answer if you enjoyed this
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