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Circles | Circles - Exercise 10.5

Question 1

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.

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  • Solution

  • Transcript

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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subject-cta

Question 1

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.

  • Solution

  • Transcript

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 "

Our top 5% students will be awarded a special scholarship to Lido.

subject-cta
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