A circle has a center at (3 ,5 )(3,5) and passes through (0 ,2 )(0,2). What is the length of an arc covering (3pi ) /4 3π4 radians on the circle?

1 Answer
Massimiliano
Mar 10, 2016

l_a=9/4sqrt2pila=94√2π.

Explanation:

The radius of the circle with center in C(3,5)C(3,5) that passes through A(0,2)A(0,2) is the distance between the two points:

r=sqrt((x_A-x_C)^2+(y_A-y_C)^2)=r=√(xA−xC)2+(yA−yC)2=

=sqrt((0-3)^2+(2-5)^2)=sqrt(9+9)=sqrt(2*9)=3sqrt2=√(0−3)2+(2−5)2=√9+9=√2⋅9=3√2.

The angle at the center of a circle is, obviously, 2pi2π, so the angle of 3/4pi34π is: (3/4pi)/(2pi)=3/834π2π=38 of the whole circle.

Since the lenght of a circle is: l_c=2pir=2pi*3sqrt2=6sqrt2pilc=2πr=2π⋅3√2=6√2π, then the lenght of the arc is:

l_a=6sqrt2pi*3/8=9/4sqrt2pila=6√2π⋅38=94√2π.

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