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 $\frac{3 π}{4}$ $(3pi ) /4$ radians on the circle?

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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 $\frac{3 π}{4}$ $(3pi ) /4$ radians on the circle?

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Answer 1 · 2016-03-10T13:48:26

$l_{a} = \frac{9}{4} \sqrt{2} π$ $l_a=9/4sqrt2pi$ .

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:

$r = \sqrt{{(x_{A} - x_{C})}_{A}^{2} + {(y_{A} - y_{C})}_{A}^{2}} =$ $r=sqrt((x_A-x_C)^2+(y_A-y_C)^2)=$

$= \sqrt{{(0 - 3)}^{2} + {(2 - 5)}^{2}} = \sqrt{9 + 9} = \sqrt{2 \cdot 9} = 3 \sqrt{2}$ $=sqrt((0-3)^2+(2-5)^2)=sqrt(9+9)=sqrt(2*9)=3sqrt2$ .

The angle at the center of a circle is, obviously, $2 π$ $2pi$ , so the angle of $\frac{3}{4} π$ $3/4pi$ is: $\frac{\frac{3}{4} π}{2 π} = \frac{3}{8}$ $(3/4pi)/(2pi)=3/8$ of the whole circle.

Since the lenght of a circle is: $l_{c} = 2 π r = 2 π \cdot 3 \sqrt{2} = 6 \sqrt{2} π$ $l_c=2pir=2pi*3sqrt2=6sqrt2pi$ , then the lenght of the arc is:

$l_{a} = 6 \sqrt{2} π \cdot \frac{3}{8} = \frac{9}{4} \sqrt{2} π$ $l_a=6sqrt2pi*3/8=9/4sqrt2pi$ .

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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 $\frac{3 π}{4}$ $(3pi ) /4$ radians on the circle?

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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 $\frac{3 π}{4}$ $(3pi ) /4$ radians on the circle?