Answer: By calculating the perimeter, we can verify if it is greater than 47.5 cm.
Step-by-step explanation:
To find the perimeter of the sector, we need to add the lengths of two parts: the curved part (arc) and the straight part (radii).
1. Calculating the length of the arc:
The formula for the length of an arc in a sector is given by: (angle/360) * (2 * pi * radius)
In this case, the radius is given as 11 cm. To find the angle, we can use the fact that the sector is 135° out of a full circle (360°).
So, the length of the arc is:
(135/360) * (2 * pi * 11) = (3/8) * (2 * pi * 11)
2. Calculating the length of the radii:
Since the sector is a part of a circle, the radii form two sides of a triangle. We can use the Pythagorean theorem to find the length of the radii.
The hypotenuse of the triangle is the radius of the circle, which is 11 cm. The other two sides are the lengths of the radii.
Let's assume the length of one radius is x cm. Then, using the Pythagorean theorem:
x^2 + x^2 = 11^2
2x^2 = 121
x^2 = 121/2
x = sqrt(121/2)
3. Calculating the perimeter:
Now that we have the length of the arc and the lengths of the radii, we can find the perimeter by adding these lengths together.
Perimeter = length of the arc + 2 * length of the radii
Substituting the values we calculated:
Perimeter = (3/8) * (2 * pi * 11) + 2 * sqrt(121/2)
Simplifying this expression will give us the exact perimeter of the sector.
By calculating the perimeter, we can verify if it is greater than 47.5 cm.
