A circle has a chord that goes from $\frac{π}{2}$ $( pi)/2$ to $\frac{13 π}{6}$ $(13 pi) / 6$ radians on the circle. If the area of the circle is $5 π$ $5 pi$ , what is the length of the chord?

Question

A circle has a chord that goes from $\frac{π}{2}$ $( pi)/2$ to $\frac{13 π}{6}$ $(13 pi) / 6$ radians on the circle. If the area of the circle is $5 π$ $5 pi$ , what is the length of the chord?

1 Answer

Impact of this question

2045 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-06-27T14:57:35

The length of the chord is $\sqrt{5}$ $sqrt5$ .

Explanation:

Let the center of the circle be point $O$ $O$ . Let the point at $\frac{π}{2}$ $pi/2$ be point $A$ $A$ and let the point at $\frac{13 π}{6}$ $(13pi)/6$ be point $B$ $B$ . Since $\frac{13 π}{6} = 2 π + \frac{π}{6}$ $(13pi)/6=2pi+pi/6$ , point $B$ $B$ can be thought of as located at an angle of $\frac{π}{6}$ $pi/6$ . Therefore, $m ∠ A O B = \frac{π}{2} - \frac{π}{6} = \frac{π}{3}$ $m/_AOB=pi/2-pi/6=pi/3$ . Since points $A$ $A$ and $B$ $B$ are located on the circle, $O A = O B$ $OA=OB$ because they are both radii. Since $O A = O B$ $OA=OB$ , we know that $m ∠ O A B = m ∠ O B A$ $m/_OAB=m/_OBA$ and since $m ∠ A O B = \frac{π}{3}$ $m/_AOB=pi/3$ , we know that $m ∠ O A B = m ∠ O B A = m ∠ A O B = \frac{π}{3}$ $m/_OAB=m/_OBA=m/_AOB=pi/3$ and that $O A = O B = A B$ $OA=OB=AB$ .

The area of a circle is given by $A = π r^{2}$ $A=pir^2$ so we know $5 π = π r^{2}$ $5pi=pir^2$ or that $r = \sqrt{5}$ $r=sqrt5$ . Since $O A = r = \sqrt{5}$ $OA=r=sqrt5$ and chord $A B = O A$ $AB=OA$ , $A B = \sqrt{5}$ $AB=sqrt5$ .

Science

Math

Humanities

... and beyond

A circle has a chord that goes from $\frac{π}{2}$ $( pi)/2$ to $\frac{13 π}{6}$ $(13 pi) / 6$ radians on the circle. If the area of the circle is $5 π$ $5 pi$ , what is the length of the chord?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

A circle has a chord that goes from π2( pi)/2 to 13π6(13 pi) / 6 radians on the circle. If the area of the circle is 5π5 pi , what is the length of the chord?

1 Answer

Explanation:

Related questions

Impact of this question

A circle has a chord that goes from $\frac{π}{2}$ $( pi)/2$ to $\frac{13 π}{6}$ $(13 pi) / 6$ radians on the circle. If the area of the circle is $5 π$ $5 pi$ , what is the length of the chord?