The Given figure is a sector of a Circle with Centre at point O

but perimeter of rectangle is 26 means we have three possibilities for length & breath i.e. (9,4), (8,5) or (7,6)

The Highlighted part which you mentioned in your previous post has posed two mistakes

1) Nowhere it is given that sides of the rectangle have Integer Length

2) You have Missed the fact that Both Diagonal of Rectangle are always equal and when you draw the second Diagonal OB then it will be same as Radius of the sector = 10 and AC = OB = 10

The Remaining Calculation is here.

We don't need the length of the sides of Rectangle as half the perimeter of Rectangle will be equal to (Length + width) of Rectangle = 26/2 = 13
and AY + CX = (OY + OX) - (OA+OC) = (10+10) - (Length+Width of rectangle) = 20 - 13 = 7

I hope it answers your query!

Got it

My mistake

Thanks a lot

Jimmy

Prompt analysis
Let radius be R, OA = aOC =b
Four segments are there AY = R-a, AC =R, XY = /2 x R, CY =R - b and a2 + b2 = R2

Superset
The value of the perimeter = AY + AC + CY + XY will be a real number.

Translation
In order to find the value of perimeter, we need:
1# exact value of three variables.
2# 3 equations to solve 3 variables.

Statement analysis

St 1: R = 10 and a^2 + b^2 = R^2. 3 variables, 2 equation. INSUFFICIENT as we don't have any idea about the a,b.
St 2: 2(a+b) = 26 or a +b =13 and a^2 + b^2 = R^2. 3 variables, 2 equation. INSUFFICIENT as we don't have any idea about the a,b and R.

St1 & St 2: R = 10, a +b =13 and a^2 + b^2 = R^2. 3 variable and 3 equations. SUFFICIENT.

Option C

This is how I visualized the problem, we have an arc with center o. The perimeter of the arc can be found using the formula, 2Πr c/360. We need to calculate the perimeter of triangle and subtract it from the arc perimeter.

1) Statement 1 is insufficient as we cannot calculate the perimeter of the triangle.
2) Statement 2 has the perimeter of the rectangle. The area of each triangle inside perimeter will be half of the rectangle. However, this statement does not mention about arc. Insufficient.

Combining both the above statements, we can calculate the perimeter of arc and the triangle (perimeter / 2). C

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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