Question

Find the length of a circle if the circumference is 32 and The measure of the angle ACB equals 90° find the length of AB

Answer 1

the arc length of AB is 8

Step-by-step explanation

the length of arc is given by:

`\begin{gathered} arc=(\theta)/(360)\cdot2\pi r \\ when\text{ }\theta\text{ is measured in degr}ees \end{gathered}`

and the circumference of a circle is given by

`\begin{gathered} C=2\pi r \\ \text{if r (radius)is isolated} \\ (C)/(2\pi)=r \\ \end{gathered}`

so

Step 1

find the radius of the circle

`\begin{gathered} let \\ \text{Circumference}=\text{ C=32} \\ \text{now , replace in the equation that relates C and r} \\ (C)/(2\pi)=r \\ r=(C)/(2\pi) \\ r=(32)/(2\pi) \\ r=5.09 \\ \text{hence, the radius of the circle is 5.09 } \end{gathered}`

Step 2

Now, let's find the arc length

let

`\begin{gathered} \text{radius}=\text{ 5.09} \\ \text{angle}=90\text{ \degree } \\ \end{gathered}`

now, let's replace in teh formula for the arc length

`\begin{gathered} arc=(\theta)/(360)\cdot2\pi r \\ arc=(90)/(360)\cdot2\pi\cdot5.09 \\ \text{arc l=}(1)/(4)\cdot10.18\pi \\ \text{arc l=}8 \end{gathered}`

therefore, the arc length of AB is 8

I hope this helps you