Question

In this picture the measure of angle theta is 1 radian

Answer 1

True.

Explanation:

Answer 2

To solve this, we are going to use the formula for the area of the sector of a circle:
`A= (1)/(2) r^2 \alpha`
where

`A` is the area of the circular sector.

`r` is the radius of the circle.

`\alpha` is the central angle in radians.

We know form our problem that that the measure of the central angle is 1 radian, so
`\alpha =1`. We can also infer from the picture that the radius of the circle is 3in, so
`r=3in`. Lets replace those values in our formula to find
`A`:

`A= (1)/(2) r^2 \alpha`

`A= (1)/(2) (3in)^2(1)`

`A=4.5in^2`

We can conclude that the area of the circular sector in the picture is 4.5 square inches.

To prove that the arc length is indeed 3 inches, we are going to use the formula:
`A_(L)=r \alpha`
where

`A_(L)` is the arc length.

`r` us the radius of the circle.

`\alpha` is the central angle.

We know from our problem that
`r=3in`, and
`\alpha =1`, so lets replace those values in our formula:

`A_(L)=r \alpha`

`A_(L)=(3in) \alpha`

`A_(L)=3in`

We can conclude that the length of the arc is indeed 3 inches.