GMATinsight wrote:

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The Given figure is a sector of a Circle with Centre at point O and Radius OX and OY such that OX and OY are making 90 degrees at point O. If OABC is a rectangle then Find the Perimeter of Red Outlined part (i.e. Perimeter of AYBXCA)?

(1) Radius of the Circle is 10 cm

(2) Perimeter of Rectangle OABC = 26

Assume that \(r\) is the radius of the circle, \(a\) is the length of OA and \(c\) is the length of OC.

The perimeter of AYBXCA is AY + YB + BX + XC + CA.

Since AY = \(( r - a )\), YB + BX = \(\frac{2pi*r}{4}\), XC =\(( r - c )\) and CA = OB = \(r\), the perimeter AYBXCA is \(3r - ( a + c ) + \frac{pi * r}{2}\).

Thus, the question asks what the value of \(3r - ( a + c ) + \frac{pi * r}{2}\) is.

In addition, we have \(a^2 + c^2 = r^2\) from Pythagras' theorem.

Now, we have 3 variables ( \(a\), \(c\), and \(r\) ) and 1 equation ( \(a^2 + c^2 = r^2\) ), and so the difference of them is 2.

Hence the choice C is most likely the correct one.

We can translate the conditions to the mathematical expressions with equations as follows.

(1) \(r = 10\)

(2) \(2(a+b) = 26\) or \(a + b = 13\)

It is clear each condition alone is not sufficient.

Let's consider both conditions together.

The perimeter AYBXCA,

\(3r - ( a + c ) + \frac{pi * r}{2} = 30 - 13 + \frac{10 pi}{2} = 17 + 5 pi\)

Therefore, C is the correct answer as expected.

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