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Question 1 Circles - Exercise 10.5

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Answer:

Here, the chord AB is equal to the radius of the circle. In the above diagram, OA and OB are the

two radii of the circle.

Now, consider the ΔOAB. Here,

AB = OA = OB = radius of the circle.

So, it can be said that ΔOAB has all equal sides and thus, it is an equilateral triangle.

∴∠AOC = 60°

And, ∠ACB = 1/2 ∠AOB

So, ∠ACB = 1/2 × 60° = 30°

Now, since ACBD is a cyclic quadrilateral,

∠ADB + ∠ACB = 180° (Since they are the opposite angles of a cyclic quadrilateral)

So, ∠ADB = 180° - 30° = 150°

So, the angle subtended by the chord at a point on the minor arc and also at a point on the major arc are 150° and 30° respectively.

"hello students welcome to lido q a video
session i am self your maths tutor and
question for today is
the chord of a circle is equal to the
radius of the circle
find the angle subtended by the chord at
a point on the
minor arc and also the point on
the major r here chord a b
is equal to the radius of the circle so
in the diagram given below
o a and o b are the two radii of the
circle
on a b will be equal to
o a equal to ob
because
chord equal to radii
now consider triangle oab
in triangle oab you can see that the
three sides
a b so this is
a b and then you have
oa and ob
these three are the equal because
here a b is the chord
o a is the radius and ob is also the
radius
so this triangle is an equilateral
triangle
so we can say from
above result
triangle o a b
is equilateral
now as this triangle is equilateral
angle aoc is 60 degree
therefore angle aoc
will be equal to 60 degree
and you can see here that angle acb is
half of angle aob
so angle acb is equal to
half of angle aov
now acb will be equal to half of angle
aob and aov
60
so angle acb will be equal to 30 degree
as angle acb is 30 degree we have to
consider
now another thing that is cyclic
quadrilateral
a c b d you can see in this cyclic
quadrilateral a
c b d we have
angle adb plus angle acd
is equal to 180 degree because they are
the opposite angles of the cyclic
quadrature
so in cyclic quadrilateral
acbd
angle adb
plus angle acb
will be equal to 180 degree reason is
opposite angles of the cyclic
quadrilateral
so angle adb will be equal to we need to
find angle adb
now so angle adb will be equal to
180 minus 30.
acb is 30 remember
so 180 minus 30 is 150
and hence your measure of angle adb
is 150 degree
so the angle subtended by the cord at
the point on the minor arc and also the
point on the major r
are 150 degree and 30 degree
respectively
acb is the angle subtended by the chord
at the point on the major arc and adv
is the angle which is subtended by the
chord at the point
on the minor arc
so we will write some statement over
here
hence what we can conclude from this is
angle subtended by chord
at point on
minor arc
is 150 degree and
on major r
is 30 degree
so here angle acb
this is the 30 degree angle and
angle adb this is the 150 degree angle
if you have any queries you can drop it
in our comment section
and subscribe to lido for more such q a
thank you for watching
"

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