the diagram shows a sector of a circle, center O, radius 5r the length of the arc AB 4r. find the area of the sector in terms of r , giving your answer in its simplest form

the area of the sector in terms of r is

Given :

From the given diagram , the radius of the circle is 5r  and length of arc AB is 4r

Lets find out the central angle using length of arc formula

length of arc =

r=5r  and length = 4r

Now we replace this angle in area of sector formula

Area of sector =

So, the area of the sector in terms of r is

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10r²

Step-by-step explanation:

The following data were obtained from the question:

Radius (r) = 5r

Length of arc (L) = 4r

Area of sector (A) =?

Next, we shall determine the angle θ sustained at the centre.

Recall:

Length of arc (L) = θ/360 × 2πr

With the above formula, we shall determine the angle θ sustained at the centre as follow:

Radius (r) = 5r

Length of arc (L) = 4r

Angle at the centre θ =?

L= θ/360 × 2πr

4r = θ/360 × 2π × 5r

4r = (θ × 10πr)/360

Cross multiply

θ × 10πr = 4r × 360

Divide both side by 10πr

θ = (4r × 360) /10πr

θ = 144/π

Finally, we shall determine the area of the sector as follow:

Angle at the centre θ = 144/π

Radius (r) = 5r

Area of sector (A) =?

Area of sector (A) = θ/360 × πr²

A = (144/π)/360 × π(5r)²

A = 144/360π × π × 25r²

A = 144/360 × 25r²

A = 0.4 × 25r²

A = 10r²

Therefore, the area of the sector is 10r².

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