Question

explain why the arc length, s, divided by the radius is the same, regardless of the circles radius. move words to the lines and expressions to the boxes to complete the statements.

there are
\b radians in a circle, so the ratio of the
\b to the entire circle is given by the expression
\b. the circumference of a circle with radius r is given by the expression
\b. the arc length, s, is given by the expression
\b. dividing the expression for arc length by r gives the expression
\b. thus, the arc length divided by the radius is always equal to the measure of the
\b, regardless of the radius.

$2\pi$ $\pi r^2$ $2\pi r$ $\theta$ $\frac{\theta}{2\pi}$ $\frac{\theta}{2\pi r}$ $2\pi r \times \frac{\theta}{2\pi}$ $\pi r^2 \times \frac{\theta}{2\pi r}$ area arc length central angle

Explanation:

To solve this, we analyze each blank based on circle and radian concepts:

Step 1: Radians in a circle

A full circle has \( 2\pi \) radians. So the first blank is \( 2\pi \).

Step 2: Ratio of arc length to circle

The ratio of the arc length to the entire circle (circumference) is given by \( \frac{\theta}{2\pi} \) (since the central angle \( \theta \) (in radians) relates to the full circle \( 2\pi \)). So the second blank is "arc length", third is \( \frac{\theta}{2\pi} \).

Step 3: Circumference of a circle

Circumference of a circle with radius \( r \) is \( 2\pi r \). So fourth blank is \( 2\pi r \).

Step 4: Arc length formula

Arc length \( s \) is proportional to the central angle: \( s = 2\pi r \times \frac{\theta}{2\pi} \) (scaling circumference by the fraction of the circle the arc covers). So fifth blank is \( 2\pi r \times \frac{\theta}{2\pi} \).

Step 5: Dividing arc length by \( r \)

Divide \( s = 2\pi r \times \frac{\theta}{2\pi} \) by \( r \): \( \frac{s}{r} = \theta \). So sixth blank is \( \theta \).

Step 6: Final relationship

The arc length divided by radius equals the central angle (in radians), so seventh blank is "central angle".

Filled Statements:

There are \( \boldsymbol{2\pi} \) radians in a circle, so the ratio of the \( \boldsymbol{\text{arc length}} \) to the entire circle is given by the expression \( \boldsymbol{\frac{\theta}{2\pi}} \). The circumference of a circle with radius \( r \) is given by the expression \( \boldsymbol{2\pi r} \). The arc length, \( s \), is given by the expression \( \boldsymbol{2\pi r \times \frac{\theta}{2\pi}} \). Dividing the expression for arc length by \( r \) gives the expression \( \boldsymbol{\theta} \). Thus, the arc length divided by the radius is always equal to the measure of the \( \boldsymbol{\text{central angle}} \), regardless of the radius.

Answer:

To solve this, we analyze each blank based on circle and radian concepts:

Step 1: Radians in a circle

A full circle has \( 2\pi \) radians. So the first blank is \( 2\pi \).

Step 2: Ratio of arc length to circle

The ratio of the arc length to the entire circle (circumference) is given by \( \frac{\theta}{2\pi} \) (since the central angle \( \theta \) (in radians) relates to the full circle \( 2\pi \)). So the second blank is "arc length", third is \( \frac{\theta}{2\pi} \).

Step 3: Circumference of a circle

Circumference of a circle with radius \( r \) is \( 2\pi r \). So fourth blank is \( 2\pi r \).

Step 4: Arc length formula

Arc length \( s \) is proportional to the central angle: \( s = 2\pi r \times \frac{\theta}{2\pi} \) (scaling circumference by the fraction of the circle the arc covers). So fifth blank is \( 2\pi r \times \frac{\theta}{2\pi} \).

Step 5: Dividing arc length by \( r \)

Divide \( s = 2\pi r \times \frac{\theta}{2\pi} \) by \( r \): \( \frac{s}{r} = \theta \). So sixth blank is \( \theta \).

Step 6: Final relationship

The arc length divided by radius equals the central angle (in radians), so seventh blank is "central angle".

Filled Statements:

There are \( \boldsymbol{2\pi} \) radians in a circle, so the ratio of the \( \boldsymbol{\text{arc length}} \) to the entire circle is given by the expression \( \boldsymbol{\frac{\theta}{2\pi}} \). The circumference of a circle with radius \( r \) is given by the expression \( \boldsymbol{2\pi r} \). The arc length, \( s \), is given by the expression \( \boldsymbol{2\pi r \times \frac{\theta}{2\pi}} \). Dividing the expression for arc length by \( r \) gives the expression \( \boldsymbol{\theta} \). Thus, the arc length divided by the radius is always equal to the measure of the \( \boldsymbol{\text{central angle}} \), regardless of the radius.