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glamrockqueen7

Answered question

2021-08-09

Given: Concentric circles with radii of lengths R and r, with R > r
Prove: $A_{r i n g} = π {(B C)}^{2}$

Answer & Explanation

Ayesha Gomez

Skilled2021-08-10Added 104 answers

Given that:

Vasquez

Expert2021-12-27Added 669 answers

Step 1

From the figure -

Radius of outer circle= OC=R

Radius of inner circle= OB=r

In triangle OBC, using pythagorus theorem $O C^{2} = O B^{2} + B C^{2}$

$R^{2} = r^{2} + B C^{2}$

$B C^{2} = R^{2} - r^{2} \dots (1)$

Step 2

$Area of the ring = A_{r i n g} = Area of the outer circle - Area of the inner circle$

$A_{r i n g} = π R^{2} - π r^{2}$

$A_{r i n g} = π (R^{2} - r^{2}) [using equation (1)]$

$A_{r i n g} = π (B C^{2})$

hence, proved

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