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The diagram shows a sector OAB of a circle with centre O and radius 2 - AQA - A-Level Maths Pure - Question 3 - 2020 - Paper 1

Question 3

The diagram shows a sector OAB of a circle with centre O and radius 2. The angle AOB is θ radians and the perimeter of the sector is 6. Find the value of θ. Circl... show full transcript

Worked Solution & Example Answer:The diagram shows a sector OAB of a circle with centre O and radius 2 - AQA - A-Level Maths Pure - Question 3 - 2020 - Paper 1

Step 1

Find the relationship between perimeter and angle

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Answer

The perimeter of a sector is given by the formula:

$P = r + r + L$

where ( P ) is the perimeter, ( r ) is the radius of the circle, and ( L ) is the arc length of the sector. Here, ( r = 2 ) and the perimeter ( P = 6 ). Therefore:

$6 = 2 + 2 + L$

Simplifying this, we find:

$L = 6 - 4 = 2$

Step 2

Find the arc length in terms of angle

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Answer

The arc length ( L ) can also be expressed in terms of the angle ( θ ) in radians using the formula:

$L = rθ$

Substituting the known values into this formula gives:

$2 = 2θ$

Step 3

Solve for θ

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Answer

Dividing both sides by 2:

$θ = 1$

Thus, the value of θ is 1.

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