Q) 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.

9th Class Maths – NCERT Important Questions

Ans:

Step 1:

Let’s make a diagram for better understanding of the question:

Here, AB is the chord of length R. It subtends ∠ ACB in minor Arc and subtends ∠ ADB in major Arc.

We need to find values of these 2 angles.

Step 2:

Let’s connect points A and B with center O.

Here, in Δ OAB, sides OA and OB are radii of the circle.

Side AB is the chord equal to radius (given).

∴  Δ OAB is equilateral triangle.

∴  ∠ AOB = 60 ⁰

Step 3:

We know that the Angle subtended by a chord on Arc is half of the angle subtended on center.

∴  ∠ ADB = ∠ AOB

= x 60 ⁰ = 30 ⁰

Step 4: Method 1:

Reflex angle ∠ AOB = 360 ⁰ – 60 ⁰ = 300 ⁰

∴  ∠ ACB = (Reflex ∠ AOB)

= x 300 ⁰ = 150 ⁰

Step 4: Method 2:

Since opposite angles in a cyclical Quadrilateral are supplementary,

and ADBC is a cyclical Quadrilateral

∴  ∠ ACB + ∠ ADB = 180 ⁰ 

∴  ∠ ACB + 30 = 180 ⁰ 

∴  ∠ ACB = 180 ⁰ – 30 ⁰ = 150 ⁰

Therefore, the angle subtended by the chord at a point on the minor arc is 150 ⁰ and on the major arc is 30 ⁰

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