Bunuel wrote:

In the circle above, AC and BD are the diameters of length 10. The sum of the lengths of the arcs AXD and BYC is

(A) 4π

(B) 8π

(C) 9π

(D) 10π

(E) 16π

Attachment:

2017-11-22_1023_001.png

The key is to find the areas of

sectors AXD and BYC, combined, as a fraction of the circle.

That fraction will hold for arc lengths and circumference. Arc lengths of arcs AXD and BYC will be the same fraction of the circumference as the sectors are of the circle's area.

Central angles of arcs AXD and BYCThe smaller sectors' central angles are 36° each (they are vertical angles, hence equal).

Each small sector's central angle lies on a straight line with one larger sector's central angle. Larger sector AXD's central angle, e.g., therefore is

180° - 36° = 144°

Sectors AXD and BYC have vertical central angles. Both sectors' central angles hence = 144°. Together, their central angles total 288°.

Central angles determine fractional area. In other words, the central angles, summed, determine what fraction of the circle the sectors AXD and BYC are.

Sectors AXD and BYC make up what fraction of the circle?\(\frac{Sectors'Angles}{360°}=\frac{Sectors'Area}{CircleArea}\)

\(\frac{288}{360}=\frac{4}{5}=\frac{Part}{Whole}\)Sectors AXD and BYC make up \(\frac{4}{5}\) of the circle. Their arc lengths will be \(\frac{4}{5}\) of the circumference.

Radius of circle with diameter of 10:

\(10 = 2r\), \(r = 5\)

Circumference: \(2πr = 10π\)

Arc lengths AXD and BYC are \(\frac{4}{5}\) of circumference:

\(\frac{4}{5} * 10π = 8π\)

Answer B

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